# Quiz 3: Functions

Question

## Question 1

What is the largest possible domain and the corresponding range of the following function?
$f\left(x\right)=ln\left(x+2\right)$
 a) Domain : $ℝ$, Range : $ℝ$ b) Domain : $\left(2,\infty \right)$, Range : $ℝ$ c) Domain : $\left(-2,\infty \right)$, Range : $\left[0,\infty \right)$ d) Domain : $\left[-2,\infty \right)$, Range : $\left[0,\infty \right)$ e) Domain : $\left(-2,\infty \right)$, Range : $ℝ$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

## Question 2

What is the largest possible domain and the corresponding range of the function
$f\left(x\right)=sin\left({e}^{x}\right)\phantom{\rule{1em}{0ex}}?$
 a) Domain : $\left(0,\infty \right)$, Range : $\left[-1,1\right]$ b) Domain : $ℝ$, Range : $\left[-1,1\right]$ c) Domain : $ℝ$, Range : $ℝ$ d) Domain : $\left(0,\infty \right)$, Range : $ℝ$ e) Domain : $\left[0,2\pi \right]$, Range : $\left[-1,1\right]$

Not correct. Choice (a) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 3

If $f\left(x\right)={x}^{2}$ and $g\left(x\right)=x+1$ then the composite function $\left(f\circ g\right)\left(x\right)$ is equal to
 a) ${x}^{2}+1$ b) $\left({x}^{2}+1\right)+1$ c) ${\left({x}^{2}+1\right)}^{2}$ d) ${\left(x+1\right)}^{2}$ e) $\sqrt{x+1}$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (e) is false.

## Question 4

If $f\left(x\right)=\sqrt{x}$, $g\left(x\right)=x+1$ and $h\left(x\right)={e}^{x}$ then $\left(\rightf\circ g\circ h\left)\right\left(x\right)$ is given by
 a) $\sqrt{{e}^{x+1}}$ b) $\sqrt{{e}^{x}+1}$ c) ${e}^{\frac{x}{2}}+1$ d) ${e}^{\left(\sqrt{x}+1\right)}$ e) $\sqrt{{e}^{x}}+1$

Not correct. Choice (a) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 5

The notation $f:A\to B$ means
 a) the function $f$ takes all elements of set $A$ as inputs and produces all elements of set $B$ as outputs. b) the function $f$ takes all elements of set $A$ as inputs and the outputs will always be in set $B$. c) the function $f$ takes some subset of $A$ as inputs and the outputs will always be in set $B$. d) the domain of the function $f$ is $A$ and its range is $B$. e) none of the above.

Not correct. Choice (a) is false.
$B$ is the codomain of $f$, so the range of $f$ is a subset of $B$. The range is not necessarily equal to $B$.
$f:A\to B$’ means that the function is properly defined on all elements of $A$ and that its input values come from $A$. Its output values all lie in $B$ ( the target set or codomain). However $B$ need not be equal to the range. The range ( the set of values which $f$ actually maps to ) is a subset of $B$ and may be equal to $B$.
Not correct. Choice (c) is false.
$A$ is equal to the domain of $f$.
Not correct. Choice (d) is false.
$B$ is the codomain of $f$, not necessarily the range.
Not correct. Choice (e) is false.

## Question 6

Consider the functions:
$f:ℝ\to ℝ;\phantom{\rule{1em}{0ex}}f\left(x\right)={x}^{4}+1$
$g:\left[0,\infty \right)\to ℝ;\phantom{\rule{1em}{0ex}}g\left(x\right)={x}^{4}+1$
$h:ℝ\to \left[1,\infty \right);\phantom{\rule{1em}{0ex}}h\left(x\right)={x}^{4}+1$
Check all statements which are true.
 a) $f$ is a surjective function. b) $g$ is a surjective function. c) $h$ is a surjective function. d) All three functions are surjective. e) $f$ and $h$ are surjective functions, and $g$ is not.

There is at least one mistake.
For example, choice (a) should be false.
$f$ is not surjective, since its range is $\left[1,\infty \right)$, not $ℝ$.
There is at least one mistake.
For example, choice (b) should be false.
$g$ is not surjective, since its range is $\left[1,\infty \right)$, not $ℝ$.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be false.
The only surjective function is $h$.
There is at least one mistake.
For example, choice (e) should be false.
The only surjective function is $h$.
1. False. $f$ is not surjective, since its range is $\left[1,\infty \right)$, not $ℝ$.
2. False. $g$ is not surjective, since its range is $\left[1,\infty \right)$, not $ℝ$.
3. True.
4. False. The only surjective function is $h$.
5. False. The only surjective function is $h$.

## Question 7

Consider the functions:
$f:ℝ\to ℝ;\phantom{\rule{1em}{0ex}}f\left(x\right)={e}^{x}$
$g:ℝ\to \left[1,\infty \right);\phantom{\rule{1em}{0ex}}g\left(x\right)={x}^{4}+1$
$h:\left[0,\infty \right)\to ℝ;\phantom{\rule{1em}{0ex}}h\left(x\right)={x}^{4}+1$
Which of the following statements are true?
 a) $f$ is an injective function. b) $f$ is surjective. c) $g$ is an injective function. d) $g$ is surjective. e) $h$ is an injective function. f) $h$ is surjective.

There is at least one mistake.
For example, choice (a) should be true.
If ${e}^{a}={e}^{b}$ then $a=b$, so $f$ is injective.
There is at least one mistake.
For example, choice (b) should be false.
The range of $f$ is $\left(0,\infty \right)$, so $f$ is not surjective.
There is at least one mistake.
For example, choice (c) should be false.
As $g\left(-1\right)=2=g\left(1\right)$, $g$ is not injective.
There is at least one mistake.
For example, choice (d) should be true.
The range of $g$ is $\left[1,\infty \right)$, so $g$ is surjective.
There is at least one mistake.
For example, choice (e) should be true.
As ${x}^{4}+1$ is an increasing function on $\left[0,\infty \right)$, this is an injective function.
There is at least one mistake.
For example, choice (f) should be false.
The range of $h$ is $\left[1,\infty \right)$ so $h$ is not surjective.
1. True. If ${e}^{a}={e}^{b}$ then $a=b$, so $f$ is injective.
2. False. The range of $f$ is $\left(0,\infty \right)$, so $f$ is not surjective.
3. False. As $g\left(-1\right)=2=g\left(1\right)$, $g$ is not injective.
4. True. The range of $g$ is $\left[1,\infty \right)$, so $g$ is surjective.
5. True. As ${x}^{4}+1$ is an increasing function on $\left[0,\infty \right)$, this is an injective function.
6. False. The range of $h$ is $\left[1,\infty \right)$ so $h$ is not surjective.

## Question 8

Check all of the following functions which are injective.
 a) $f:\left(0,\infty \right)\to ℝ;f\left(x\right)=lnx$ b) $f:\left[0,\infty \right)\to ℝ;f\left(x\right)=coshx$ c) $f:ℝ\to ℝ;f\left(x\right)={x}^{3}+1$ d) $f:ℝ\to ℝ;f\left(x\right)=sinx$

There is at least one mistake.
For example, choice (a) should be true.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be false.
1. True.
2. True.
3. True.
4. False.

## Question 9

If $f:\left(0,\infty \right)\to ℝ;\phantom{\rule{1em}{0ex}}f\left(x\right)={x}^{2}+1$, and $g:ℝ\to \left(0,\infty \right);\phantom{\rule{1em}{0ex}}g\left(x\right)={e}^{x},$ find formulas for the composite function $f\circ g$ and the inverse function of $f$.
 a) $\left(f\circ g\right)\left(x\right)={e}^{2x}+1$ and ${f}^{-1}\left(x\right)=\sqrt{x-1}$ b) $\left(f\circ g\right)\left(x\right)={e}^{2x}+1$ and ${f}^{-1}\left(x\right)=-\sqrt{x-1}$ c) $\left(f\circ g\right)\left(x\right)={e}^{2x}+1$ and ${f}^{-1}\left(x\right)=±\sqrt{x-1}$ d) $\left(f\circ g\right)\left(x\right)={e}^{{x}^{2}}+1$ and ${f}^{-1}\left(x\right)=\sqrt{x-1}$ e) $\left(f\circ g\right)\left(x\right)={e}^{{x}^{2}}+1$ and ${f}^{-1}\left(x\right)=-\sqrt{x-1}$

Note that $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 10

Consider the function $f\left(x\right)=\sqrt{3+4x}$ on its natural domain. Find the inverse function ${f}^{-1}$, giving its domain and range.
 a) ${f}^{-1}:ℝ\to ℝ$ and ${f}^{-1}\left(x\right)=\frac{{x}^{2}-3}{4}$ b) ${f}^{-1}:ℝ\to ℝ$ and ${f}^{-1}\left(x\right)=\frac{1}{\sqrt{3+4x}}$ c) ${f}^{-1}:\left[0,\infty \right)\to \left[-3∕4,\infty \right)$ and ${f}^{-1}\left(x\right)=\frac{{x}^{2}-3}{4}$ d) ${f}^{-1}:ℝ\to \left[-3∕4,\infty \right)$ and ${f}^{-1}\left(x\right)=\frac{{x}^{2}-3}{4}$ e) ${f}^{-1}:\left[0,\infty \right)\to ℝ$ and ${f}^{-1}\left(x\right)=\frac{{x}^{2}-3}{4}$ f) ${f}^{-1}:\left[0,\infty \right)\to ℝ$ and ${f}^{-1}\left(x\right)=\frac{1}{\sqrt{3+4x}}$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.