# Quiz 3: Functions

## Question 1

What is the largest possible domain and the corresponding range of the following
function?

$$f\left(x\right)=ln\left(x+2\right)$$

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

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

**Not correct. Choice (c) is false.**

**Not correct. Choice (d) is false.**

**Your answer is correct.**

## Question 2

What is the largest possible domain and the corresponding range of the
function

$$f\left(x\right)=sin\left({e}^{x}\right)\phantom{\rule{1em}{0ex}}?$$

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

**Your answer is correct.**

**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

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

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

**Not correct. Choice (c) is false.**

**Your answer is correct.**

**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
$\left(\right.f\circ g\circ h\left)\right.\left(x\right)$ is given
by

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

**Your answer is correct.**

**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

**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 $B$.

**Your answer is correct.**

‘$f:A\to B$’
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:

$$f:\mathbb{R}\to \mathbb{R};\phantom{\rule{1em}{0ex}}f\left(x\right)={x}^{4}+1$$

$$g:\left[0,\infty \right)\to \mathbb{R};\phantom{\rule{1em}{0ex}}g\left(x\right)={x}^{4}+1$$

$$h:\mathbb{R}\to \left[1,\infty \right);\phantom{\rule{1em}{0ex}}h\left(x\right)={x}^{4}+1$$

Check all statements which are true.
**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 $\mathbb{R}$.

**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 $\mathbb{R}$.

**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 $h$.

**Your answers are correct**

**False**. $f$ is not surjective, since its range is $\left[1,\infty \right)$, not $\mathbb{R}$.**False**. $g$ is not surjective, since its range is $\left[1,\infty \right)$, not $\mathbb{R}$.**True**.**False**. The only surjective function is $h$.**False**. The only surjective function is $h$.

## Question 7

Consider the functions:

$$f:\mathbb{R}\to \mathbb{R};\phantom{\rule{1em}{0ex}}f\left(x\right)={e}^{x}$$

$$g:\mathbb{R}\to \left[1,\infty \right);\phantom{\rule{1em}{0ex}}g\left(x\right)={x}^{4}+1$$

$$h:\left[0,\infty \right)\to \mathbb{R};\phantom{\rule{1em}{0ex}}h\left(x\right)={x}^{4}+1$$

Which of the following statements are true?
**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.

**Your answers are correct**

**True**. If ${e}^{a}={e}^{b}$ then $a=b$, so $f$ is injective.**False**. The range of $f$ is $\left(0,\infty \right)$, so $f$ is not surjective.**False**. As $g\left(-1\right)=2=g\left(1\right)$, $g$ is not injective.**True**. The range of $g$ is $\left[1,\infty \right)$, so $g$ is surjective.**True**. As ${x}^{4}+1$ is an increasing function on $\left[0,\infty \right)$, this is an injective function.**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.

**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.

**Your answers are correct**

**True**.**True**.**True**.**False**.

## Question 9

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

**Your answer is correct.**

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.

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

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

**Your answer is correct.**

**Not correct. Choice (d) is false.**

**Not correct. Choice (e) is false.**

**Not correct. Choice (f) is false.**