Prove that $f(x)$ is bijective?
$begingroup$
We have this function: $f(x)=2x+1$ where $f:mathbb{R} to mathbb{R}$.
How can I prove that it is bijective?
I still visit school so an explanation for each step would be appreciated.
I know what injective and surjective means and I know that I have to prove both in separate steps, but I don't really understand how this should be done.
functions proof-explanation
$endgroup$
add a comment |
$begingroup$
We have this function: $f(x)=2x+1$ where $f:mathbb{R} to mathbb{R}$.
How can I prove that it is bijective?
I still visit school so an explanation for each step would be appreciated.
I know what injective and surjective means and I know that I have to prove both in separate steps, but I don't really understand how this should be done.
functions proof-explanation
$endgroup$
$begingroup$
Ok, so what does "injective" mean? Write down the condition explicitly; where are you having trouble applying it?
$endgroup$
– T. Bongers
Oct 31 '17 at 18:54
$begingroup$
Injectivity: Suppose $f(a)=f(b)$. Try to show that this implies that $a=b$. Is it true that $2a+1=2b+1$ means that $a=b$? Why? Try using highschool algebra. Surjectivity: Try to show that each $y$ in the codomain has some $x$ that maps to it. Find something that you can put in the underlined space so that $f(underline{~~~~~})=y$
$endgroup$
– JMoravitz
Oct 31 '17 at 18:54
$begingroup$
1) In English it's more customary to say "I'm still in school", rather than "visit." 2) Regardless of whether you're in school or not: to prove it's injective you need to prove that $f(x) = f(y) implies x = y$, which in this case means $2x + 1 = 2y + 1 implies x = y$. To prove it's surjective means proving that if $x in mathbb{R}$, then $x = f(y)$ for some $y in mathbb{R}$, which in this case means $x = 2y + 1$. Can you show where you're stuck on these problems?
$endgroup$
– Chris
Oct 31 '17 at 18:56
$begingroup$
I know that that injective means that if we have a function one value of the codomain will hit 0-1 Values of the image. But i don't know why f(a)=f(b) so how should this help to prove that the function is injective?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:00
$begingroup$
Tip: I think you can use the kernal of the function to prove bijection, i.e. to prove injectivity and surjectivity.
$endgroup$
– Antinous
Dec 27 '18 at 16:21
add a comment |
$begingroup$
We have this function: $f(x)=2x+1$ where $f:mathbb{R} to mathbb{R}$.
How can I prove that it is bijective?
I still visit school so an explanation for each step would be appreciated.
I know what injective and surjective means and I know that I have to prove both in separate steps, but I don't really understand how this should be done.
functions proof-explanation
$endgroup$
We have this function: $f(x)=2x+1$ where $f:mathbb{R} to mathbb{R}$.
How can I prove that it is bijective?
I still visit school so an explanation for each step would be appreciated.
I know what injective and surjective means and I know that I have to prove both in separate steps, but I don't really understand how this should be done.
functions proof-explanation
functions proof-explanation
edited Dec 27 '18 at 8:24
Eevee Trainer
6,62311237
6,62311237
asked Oct 31 '17 at 18:52
Jan BrownJan Brown
16
16
$begingroup$
Ok, so what does "injective" mean? Write down the condition explicitly; where are you having trouble applying it?
$endgroup$
– T. Bongers
Oct 31 '17 at 18:54
$begingroup$
Injectivity: Suppose $f(a)=f(b)$. Try to show that this implies that $a=b$. Is it true that $2a+1=2b+1$ means that $a=b$? Why? Try using highschool algebra. Surjectivity: Try to show that each $y$ in the codomain has some $x$ that maps to it. Find something that you can put in the underlined space so that $f(underline{~~~~~})=y$
$endgroup$
– JMoravitz
Oct 31 '17 at 18:54
$begingroup$
1) In English it's more customary to say "I'm still in school", rather than "visit." 2) Regardless of whether you're in school or not: to prove it's injective you need to prove that $f(x) = f(y) implies x = y$, which in this case means $2x + 1 = 2y + 1 implies x = y$. To prove it's surjective means proving that if $x in mathbb{R}$, then $x = f(y)$ for some $y in mathbb{R}$, which in this case means $x = 2y + 1$. Can you show where you're stuck on these problems?
$endgroup$
– Chris
Oct 31 '17 at 18:56
$begingroup$
I know that that injective means that if we have a function one value of the codomain will hit 0-1 Values of the image. But i don't know why f(a)=f(b) so how should this help to prove that the function is injective?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:00
$begingroup$
Tip: I think you can use the kernal of the function to prove bijection, i.e. to prove injectivity and surjectivity.
$endgroup$
– Antinous
Dec 27 '18 at 16:21
add a comment |
$begingroup$
Ok, so what does "injective" mean? Write down the condition explicitly; where are you having trouble applying it?
$endgroup$
– T. Bongers
Oct 31 '17 at 18:54
$begingroup$
Injectivity: Suppose $f(a)=f(b)$. Try to show that this implies that $a=b$. Is it true that $2a+1=2b+1$ means that $a=b$? Why? Try using highschool algebra. Surjectivity: Try to show that each $y$ in the codomain has some $x$ that maps to it. Find something that you can put in the underlined space so that $f(underline{~~~~~})=y$
$endgroup$
– JMoravitz
Oct 31 '17 at 18:54
$begingroup$
1) In English it's more customary to say "I'm still in school", rather than "visit." 2) Regardless of whether you're in school or not: to prove it's injective you need to prove that $f(x) = f(y) implies x = y$, which in this case means $2x + 1 = 2y + 1 implies x = y$. To prove it's surjective means proving that if $x in mathbb{R}$, then $x = f(y)$ for some $y in mathbb{R}$, which in this case means $x = 2y + 1$. Can you show where you're stuck on these problems?
$endgroup$
– Chris
Oct 31 '17 at 18:56
$begingroup$
I know that that injective means that if we have a function one value of the codomain will hit 0-1 Values of the image. But i don't know why f(a)=f(b) so how should this help to prove that the function is injective?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:00
$begingroup$
Tip: I think you can use the kernal of the function to prove bijection, i.e. to prove injectivity and surjectivity.
$endgroup$
– Antinous
Dec 27 '18 at 16:21
$begingroup$
Ok, so what does "injective" mean? Write down the condition explicitly; where are you having trouble applying it?
$endgroup$
– T. Bongers
Oct 31 '17 at 18:54
$begingroup$
Ok, so what does "injective" mean? Write down the condition explicitly; where are you having trouble applying it?
$endgroup$
– T. Bongers
Oct 31 '17 at 18:54
$begingroup$
Injectivity: Suppose $f(a)=f(b)$. Try to show that this implies that $a=b$. Is it true that $2a+1=2b+1$ means that $a=b$? Why? Try using highschool algebra. Surjectivity: Try to show that each $y$ in the codomain has some $x$ that maps to it. Find something that you can put in the underlined space so that $f(underline{~~~~~})=y$
$endgroup$
– JMoravitz
Oct 31 '17 at 18:54
$begingroup$
Injectivity: Suppose $f(a)=f(b)$. Try to show that this implies that $a=b$. Is it true that $2a+1=2b+1$ means that $a=b$? Why? Try using highschool algebra. Surjectivity: Try to show that each $y$ in the codomain has some $x$ that maps to it. Find something that you can put in the underlined space so that $f(underline{~~~~~})=y$
$endgroup$
– JMoravitz
Oct 31 '17 at 18:54
$begingroup$
1) In English it's more customary to say "I'm still in school", rather than "visit." 2) Regardless of whether you're in school or not: to prove it's injective you need to prove that $f(x) = f(y) implies x = y$, which in this case means $2x + 1 = 2y + 1 implies x = y$. To prove it's surjective means proving that if $x in mathbb{R}$, then $x = f(y)$ for some $y in mathbb{R}$, which in this case means $x = 2y + 1$. Can you show where you're stuck on these problems?
$endgroup$
– Chris
Oct 31 '17 at 18:56
$begingroup$
1) In English it's more customary to say "I'm still in school", rather than "visit." 2) Regardless of whether you're in school or not: to prove it's injective you need to prove that $f(x) = f(y) implies x = y$, which in this case means $2x + 1 = 2y + 1 implies x = y$. To prove it's surjective means proving that if $x in mathbb{R}$, then $x = f(y)$ for some $y in mathbb{R}$, which in this case means $x = 2y + 1$. Can you show where you're stuck on these problems?
$endgroup$
– Chris
Oct 31 '17 at 18:56
$begingroup$
I know that that injective means that if we have a function one value of the codomain will hit 0-1 Values of the image. But i don't know why f(a)=f(b) so how should this help to prove that the function is injective?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:00
$begingroup$
I know that that injective means that if we have a function one value of the codomain will hit 0-1 Values of the image. But i don't know why f(a)=f(b) so how should this help to prove that the function is injective?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:00
$begingroup$
Tip: I think you can use the kernal of the function to prove bijection, i.e. to prove injectivity and surjectivity.
$endgroup$
– Antinous
Dec 27 '18 at 16:21
$begingroup$
Tip: I think you can use the kernal of the function to prove bijection, i.e. to prove injectivity and surjectivity.
$endgroup$
– Antinous
Dec 27 '18 at 16:21
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
As you rightly state, we need to show that $f$ is both injective and surjective.
$textbf{$f$ is injective}$
Recall the definition of an injective function: A function is injective if, whenever $f(x)=f(y)$, it must follow that $x=y$.
Now, suppose $f(x)=2x+1$ is equal to $f(y)=2y+1$, then $$2x+1=2y+1 Rightarrow x=y,$$ so $f$ is injective.
$textbf{$f$ is surjective}$
For $f$ to be surjective, it must follow that every $z in Bbb R$ can be written as $z=f(m)$ for some $m$. We observe that $m=frac{z-1}{2}$ is indeed an element of the real numbers so we can and so $f$ is surjective.
Since $f$ is both injective and surjective, by extension we have that it's bijective as desired.
$endgroup$
$begingroup$
Thank you so much but i don't know why f(x)=f(y) could you please explain why this proves that the function is injective as you did in the surjective part?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:15
$begingroup$
@JanBrown It's just the definition of what it means for a function to be injective.
$endgroup$
– thesmallprint
Oct 31 '17 at 19:20
add a comment |
$begingroup$
I want to present an alternative interpretation of what thesmallprint introduced in his answer, because - particularly the injective one - is very helpful to keep in mind.
Recall our function is $f : mathbb{R} to mathbb{R}$ defined by $f(x) = 2x + 1$.
Prove $f$ is injective
By definition, a function $f$ is injective if, whenever $f(x) = f(y)$, we have $x = y$. We can also look at this in the contrapositive since - $f$ is also injective if, whenever $x neq y$, $f(x) neq f(y)$. Sometimes, this angle of approach is easier to look at, since it is basically "unequal inputs have unequal outputs."
This also establishes the notion of injectivity as $f$ passing a horizontal line test: graph the function, and, if at any point you can place a horizontal line through the function and it crosses at $2$ or more points, then it is not injective. You should be able to easily visualize the graph of our given $f$ and confirm it, at least heuristically, in this sense.
Formally, notice that
$$x neq y ;;; Rightarrow ;;; 2x neq 2y ;;; Rightarrow ;;; 2x + 1 neq 2y + 1 ;;; Rightarrow ;;; f(x) neq f(y)$$
Thus, $f$ is injective.
Prove $f$ is surjective
Consider elements $y$ in the codomain $mathbb{R}$. What would the corresponding input $x$ under $f$? Obviously,
$$x = frac{y-1}{2}$$
To each real number $y$, then, you can associate a corresponding $x$ as the input to $f$. Thus, $f$ is surjective.
Prove $f$ is bijective
This follows by definition of bijectivity: as $f$ is injective and surjective, it is bijective.
Another notable tenet: since bijectivity implies the existence of a two-sided inverse, you could also show bijectivity by showing the existence of such an inverse. (Injectivity gives the left-sided inverse, and surjectivity the right. Given the time since the posting of this question, it has probably already been introduced to you in your coursework.) For example, let us write $f$ as
$$y = 2x + 1$$
and then solve for $x$, which yields, as before,
$$x = frac{y-1}{2}$$
Thus, if $f^{-1}(y) = frac{y-1}{2}$, where $f^{-1} : mathbb{R} to mathbb{R}$, if you show
$$(f circ f^{-1})(x) = (f^{-1} circ f)(x) = x$$
then $f$ is bijective. You can probably do this in your head.
$endgroup$
add a comment |
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2 Answers
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active
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2 Answers
2
active
oldest
votes
active
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votes
$begingroup$
As you rightly state, we need to show that $f$ is both injective and surjective.
$textbf{$f$ is injective}$
Recall the definition of an injective function: A function is injective if, whenever $f(x)=f(y)$, it must follow that $x=y$.
Now, suppose $f(x)=2x+1$ is equal to $f(y)=2y+1$, then $$2x+1=2y+1 Rightarrow x=y,$$ so $f$ is injective.
$textbf{$f$ is surjective}$
For $f$ to be surjective, it must follow that every $z in Bbb R$ can be written as $z=f(m)$ for some $m$. We observe that $m=frac{z-1}{2}$ is indeed an element of the real numbers so we can and so $f$ is surjective.
Since $f$ is both injective and surjective, by extension we have that it's bijective as desired.
$endgroup$
$begingroup$
Thank you so much but i don't know why f(x)=f(y) could you please explain why this proves that the function is injective as you did in the surjective part?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:15
$begingroup$
@JanBrown It's just the definition of what it means for a function to be injective.
$endgroup$
– thesmallprint
Oct 31 '17 at 19:20
add a comment |
$begingroup$
As you rightly state, we need to show that $f$ is both injective and surjective.
$textbf{$f$ is injective}$
Recall the definition of an injective function: A function is injective if, whenever $f(x)=f(y)$, it must follow that $x=y$.
Now, suppose $f(x)=2x+1$ is equal to $f(y)=2y+1$, then $$2x+1=2y+1 Rightarrow x=y,$$ so $f$ is injective.
$textbf{$f$ is surjective}$
For $f$ to be surjective, it must follow that every $z in Bbb R$ can be written as $z=f(m)$ for some $m$. We observe that $m=frac{z-1}{2}$ is indeed an element of the real numbers so we can and so $f$ is surjective.
Since $f$ is both injective and surjective, by extension we have that it's bijective as desired.
$endgroup$
$begingroup$
Thank you so much but i don't know why f(x)=f(y) could you please explain why this proves that the function is injective as you did in the surjective part?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:15
$begingroup$
@JanBrown It's just the definition of what it means for a function to be injective.
$endgroup$
– thesmallprint
Oct 31 '17 at 19:20
add a comment |
$begingroup$
As you rightly state, we need to show that $f$ is both injective and surjective.
$textbf{$f$ is injective}$
Recall the definition of an injective function: A function is injective if, whenever $f(x)=f(y)$, it must follow that $x=y$.
Now, suppose $f(x)=2x+1$ is equal to $f(y)=2y+1$, then $$2x+1=2y+1 Rightarrow x=y,$$ so $f$ is injective.
$textbf{$f$ is surjective}$
For $f$ to be surjective, it must follow that every $z in Bbb R$ can be written as $z=f(m)$ for some $m$. We observe that $m=frac{z-1}{2}$ is indeed an element of the real numbers so we can and so $f$ is surjective.
Since $f$ is both injective and surjective, by extension we have that it's bijective as desired.
$endgroup$
As you rightly state, we need to show that $f$ is both injective and surjective.
$textbf{$f$ is injective}$
Recall the definition of an injective function: A function is injective if, whenever $f(x)=f(y)$, it must follow that $x=y$.
Now, suppose $f(x)=2x+1$ is equal to $f(y)=2y+1$, then $$2x+1=2y+1 Rightarrow x=y,$$ so $f$ is injective.
$textbf{$f$ is surjective}$
For $f$ to be surjective, it must follow that every $z in Bbb R$ can be written as $z=f(m)$ for some $m$. We observe that $m=frac{z-1}{2}$ is indeed an element of the real numbers so we can and so $f$ is surjective.
Since $f$ is both injective and surjective, by extension we have that it's bijective as desired.
edited Oct 31 '17 at 19:20
answered Oct 31 '17 at 19:05
thesmallprintthesmallprint
2,6611618
2,6611618
$begingroup$
Thank you so much but i don't know why f(x)=f(y) could you please explain why this proves that the function is injective as you did in the surjective part?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:15
$begingroup$
@JanBrown It's just the definition of what it means for a function to be injective.
$endgroup$
– thesmallprint
Oct 31 '17 at 19:20
add a comment |
$begingroup$
Thank you so much but i don't know why f(x)=f(y) could you please explain why this proves that the function is injective as you did in the surjective part?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:15
$begingroup$
@JanBrown It's just the definition of what it means for a function to be injective.
$endgroup$
– thesmallprint
Oct 31 '17 at 19:20
$begingroup$
Thank you so much but i don't know why f(x)=f(y) could you please explain why this proves that the function is injective as you did in the surjective part?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:15
$begingroup$
Thank you so much but i don't know why f(x)=f(y) could you please explain why this proves that the function is injective as you did in the surjective part?
$endgroup$
– Jan Brown
Oct 31 '17 at 19:15
$begingroup$
@JanBrown It's just the definition of what it means for a function to be injective.
$endgroup$
– thesmallprint
Oct 31 '17 at 19:20
$begingroup$
@JanBrown It's just the definition of what it means for a function to be injective.
$endgroup$
– thesmallprint
Oct 31 '17 at 19:20
add a comment |
$begingroup$
I want to present an alternative interpretation of what thesmallprint introduced in his answer, because - particularly the injective one - is very helpful to keep in mind.
Recall our function is $f : mathbb{R} to mathbb{R}$ defined by $f(x) = 2x + 1$.
Prove $f$ is injective
By definition, a function $f$ is injective if, whenever $f(x) = f(y)$, we have $x = y$. We can also look at this in the contrapositive since - $f$ is also injective if, whenever $x neq y$, $f(x) neq f(y)$. Sometimes, this angle of approach is easier to look at, since it is basically "unequal inputs have unequal outputs."
This also establishes the notion of injectivity as $f$ passing a horizontal line test: graph the function, and, if at any point you can place a horizontal line through the function and it crosses at $2$ or more points, then it is not injective. You should be able to easily visualize the graph of our given $f$ and confirm it, at least heuristically, in this sense.
Formally, notice that
$$x neq y ;;; Rightarrow ;;; 2x neq 2y ;;; Rightarrow ;;; 2x + 1 neq 2y + 1 ;;; Rightarrow ;;; f(x) neq f(y)$$
Thus, $f$ is injective.
Prove $f$ is surjective
Consider elements $y$ in the codomain $mathbb{R}$. What would the corresponding input $x$ under $f$? Obviously,
$$x = frac{y-1}{2}$$
To each real number $y$, then, you can associate a corresponding $x$ as the input to $f$. Thus, $f$ is surjective.
Prove $f$ is bijective
This follows by definition of bijectivity: as $f$ is injective and surjective, it is bijective.
Another notable tenet: since bijectivity implies the existence of a two-sided inverse, you could also show bijectivity by showing the existence of such an inverse. (Injectivity gives the left-sided inverse, and surjectivity the right. Given the time since the posting of this question, it has probably already been introduced to you in your coursework.) For example, let us write $f$ as
$$y = 2x + 1$$
and then solve for $x$, which yields, as before,
$$x = frac{y-1}{2}$$
Thus, if $f^{-1}(y) = frac{y-1}{2}$, where $f^{-1} : mathbb{R} to mathbb{R}$, if you show
$$(f circ f^{-1})(x) = (f^{-1} circ f)(x) = x$$
then $f$ is bijective. You can probably do this in your head.
$endgroup$
add a comment |
$begingroup$
I want to present an alternative interpretation of what thesmallprint introduced in his answer, because - particularly the injective one - is very helpful to keep in mind.
Recall our function is $f : mathbb{R} to mathbb{R}$ defined by $f(x) = 2x + 1$.
Prove $f$ is injective
By definition, a function $f$ is injective if, whenever $f(x) = f(y)$, we have $x = y$. We can also look at this in the contrapositive since - $f$ is also injective if, whenever $x neq y$, $f(x) neq f(y)$. Sometimes, this angle of approach is easier to look at, since it is basically "unequal inputs have unequal outputs."
This also establishes the notion of injectivity as $f$ passing a horizontal line test: graph the function, and, if at any point you can place a horizontal line through the function and it crosses at $2$ or more points, then it is not injective. You should be able to easily visualize the graph of our given $f$ and confirm it, at least heuristically, in this sense.
Formally, notice that
$$x neq y ;;; Rightarrow ;;; 2x neq 2y ;;; Rightarrow ;;; 2x + 1 neq 2y + 1 ;;; Rightarrow ;;; f(x) neq f(y)$$
Thus, $f$ is injective.
Prove $f$ is surjective
Consider elements $y$ in the codomain $mathbb{R}$. What would the corresponding input $x$ under $f$? Obviously,
$$x = frac{y-1}{2}$$
To each real number $y$, then, you can associate a corresponding $x$ as the input to $f$. Thus, $f$ is surjective.
Prove $f$ is bijective
This follows by definition of bijectivity: as $f$ is injective and surjective, it is bijective.
Another notable tenet: since bijectivity implies the existence of a two-sided inverse, you could also show bijectivity by showing the existence of such an inverse. (Injectivity gives the left-sided inverse, and surjectivity the right. Given the time since the posting of this question, it has probably already been introduced to you in your coursework.) For example, let us write $f$ as
$$y = 2x + 1$$
and then solve for $x$, which yields, as before,
$$x = frac{y-1}{2}$$
Thus, if $f^{-1}(y) = frac{y-1}{2}$, where $f^{-1} : mathbb{R} to mathbb{R}$, if you show
$$(f circ f^{-1})(x) = (f^{-1} circ f)(x) = x$$
then $f$ is bijective. You can probably do this in your head.
$endgroup$
add a comment |
$begingroup$
I want to present an alternative interpretation of what thesmallprint introduced in his answer, because - particularly the injective one - is very helpful to keep in mind.
Recall our function is $f : mathbb{R} to mathbb{R}$ defined by $f(x) = 2x + 1$.
Prove $f$ is injective
By definition, a function $f$ is injective if, whenever $f(x) = f(y)$, we have $x = y$. We can also look at this in the contrapositive since - $f$ is also injective if, whenever $x neq y$, $f(x) neq f(y)$. Sometimes, this angle of approach is easier to look at, since it is basically "unequal inputs have unequal outputs."
This also establishes the notion of injectivity as $f$ passing a horizontal line test: graph the function, and, if at any point you can place a horizontal line through the function and it crosses at $2$ or more points, then it is not injective. You should be able to easily visualize the graph of our given $f$ and confirm it, at least heuristically, in this sense.
Formally, notice that
$$x neq y ;;; Rightarrow ;;; 2x neq 2y ;;; Rightarrow ;;; 2x + 1 neq 2y + 1 ;;; Rightarrow ;;; f(x) neq f(y)$$
Thus, $f$ is injective.
Prove $f$ is surjective
Consider elements $y$ in the codomain $mathbb{R}$. What would the corresponding input $x$ under $f$? Obviously,
$$x = frac{y-1}{2}$$
To each real number $y$, then, you can associate a corresponding $x$ as the input to $f$. Thus, $f$ is surjective.
Prove $f$ is bijective
This follows by definition of bijectivity: as $f$ is injective and surjective, it is bijective.
Another notable tenet: since bijectivity implies the existence of a two-sided inverse, you could also show bijectivity by showing the existence of such an inverse. (Injectivity gives the left-sided inverse, and surjectivity the right. Given the time since the posting of this question, it has probably already been introduced to you in your coursework.) For example, let us write $f$ as
$$y = 2x + 1$$
and then solve for $x$, which yields, as before,
$$x = frac{y-1}{2}$$
Thus, if $f^{-1}(y) = frac{y-1}{2}$, where $f^{-1} : mathbb{R} to mathbb{R}$, if you show
$$(f circ f^{-1})(x) = (f^{-1} circ f)(x) = x$$
then $f$ is bijective. You can probably do this in your head.
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I want to present an alternative interpretation of what thesmallprint introduced in his answer, because - particularly the injective one - is very helpful to keep in mind.
Recall our function is $f : mathbb{R} to mathbb{R}$ defined by $f(x) = 2x + 1$.
Prove $f$ is injective
By definition, a function $f$ is injective if, whenever $f(x) = f(y)$, we have $x = y$. We can also look at this in the contrapositive since - $f$ is also injective if, whenever $x neq y$, $f(x) neq f(y)$. Sometimes, this angle of approach is easier to look at, since it is basically "unequal inputs have unequal outputs."
This also establishes the notion of injectivity as $f$ passing a horizontal line test: graph the function, and, if at any point you can place a horizontal line through the function and it crosses at $2$ or more points, then it is not injective. You should be able to easily visualize the graph of our given $f$ and confirm it, at least heuristically, in this sense.
Formally, notice that
$$x neq y ;;; Rightarrow ;;; 2x neq 2y ;;; Rightarrow ;;; 2x + 1 neq 2y + 1 ;;; Rightarrow ;;; f(x) neq f(y)$$
Thus, $f$ is injective.
Prove $f$ is surjective
Consider elements $y$ in the codomain $mathbb{R}$. What would the corresponding input $x$ under $f$? Obviously,
$$x = frac{y-1}{2}$$
To each real number $y$, then, you can associate a corresponding $x$ as the input to $f$. Thus, $f$ is surjective.
Prove $f$ is bijective
This follows by definition of bijectivity: as $f$ is injective and surjective, it is bijective.
Another notable tenet: since bijectivity implies the existence of a two-sided inverse, you could also show bijectivity by showing the existence of such an inverse. (Injectivity gives the left-sided inverse, and surjectivity the right. Given the time since the posting of this question, it has probably already been introduced to you in your coursework.) For example, let us write $f$ as
$$y = 2x + 1$$
and then solve for $x$, which yields, as before,
$$x = frac{y-1}{2}$$
Thus, if $f^{-1}(y) = frac{y-1}{2}$, where $f^{-1} : mathbb{R} to mathbb{R}$, if you show
$$(f circ f^{-1})(x) = (f^{-1} circ f)(x) = x$$
then $f$ is bijective. You can probably do this in your head.
answered Dec 27 '18 at 8:42
Eevee TrainerEevee Trainer
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Ok, so what does "injective" mean? Write down the condition explicitly; where are you having trouble applying it?
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– T. Bongers
Oct 31 '17 at 18:54
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Injectivity: Suppose $f(a)=f(b)$. Try to show that this implies that $a=b$. Is it true that $2a+1=2b+1$ means that $a=b$? Why? Try using highschool algebra. Surjectivity: Try to show that each $y$ in the codomain has some $x$ that maps to it. Find something that you can put in the underlined space so that $f(underline{~~~~~})=y$
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– JMoravitz
Oct 31 '17 at 18:54
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1) In English it's more customary to say "I'm still in school", rather than "visit." 2) Regardless of whether you're in school or not: to prove it's injective you need to prove that $f(x) = f(y) implies x = y$, which in this case means $2x + 1 = 2y + 1 implies x = y$. To prove it's surjective means proving that if $x in mathbb{R}$, then $x = f(y)$ for some $y in mathbb{R}$, which in this case means $x = 2y + 1$. Can you show where you're stuck on these problems?
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– Chris
Oct 31 '17 at 18:56
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I know that that injective means that if we have a function one value of the codomain will hit 0-1 Values of the image. But i don't know why f(a)=f(b) so how should this help to prove that the function is injective?
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– Jan Brown
Oct 31 '17 at 19:00
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Tip: I think you can use the kernal of the function to prove bijection, i.e. to prove injectivity and surjectivity.
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– Antinous
Dec 27 '18 at 16:21