Question

Which of these functions has an inverse function? Select all that apply.
y = x
y = x^2
y = x^3
y = x^4

Answer 1

ANSWER

y=x

y=x³

EXPLANATION

A function, f(x) has an inverse if and only if

$f(a) = f(b) \Rightarrow \: a = b$

Thus, the function is one to one.

For y=x or

$f(x) = x$

$f(a) = f(b) \Rightarrow \: a = b$

Hence this function has an inverse.

For the function y=x² or f(x)=x².

$f(a) = f(b) \Rightarrow \: {a}^{2} = {b}^{2} \Rightarrow \: a = \pm \: b$

This function has no inverse on the entire real numbers.

For the function y=x³ or f(x)=x³

$f(a) = f(b) \Rightarrow \: {a}^{3} = {b}^{3} \Rightarrow \: a = b$

This function also has an inverse.

For y=x⁴ or f(x) =x⁴

$f(a) = f(b) \Rightarrow \: {a}^{4} = {b}^{4} \Rightarrow \: a = \pm \: b$

This function has no inverse over the entire real numbers.

Answer 2

Answer:

The function $x=y$ and $y=x^{3}$ are inverse function.

Step-by-step explanation:

Function inverse definition:

If a provided function f(x) is mapped x to y, then the inverse of the provided function f(x) is mapped y to x.

Or  $f(x)=f(y)\Rightarrow x=y$

Now, consider the function y = x.

Interchange the variables x and y.

$x=y$

Now, solve $x=y$ for $y$.

$x=y$

Therefore, this function has an inverse.

Consider the function $y=x^{2}$.

Interchange the variables x and y.

$x=y^{2}$

Now, solve $x=y^{2}$ for $y$.

$\pm \sqrt{x} =y$

Therefore, the function has no inverse.

Consider the function $y=x^{3}$.

Interchange the variables x and y.

$x=y^{3}$

Now, solve $x=y^{3}$ for $y$.

$\sqrt[3]{x}=y$

Therefore, the function has inverse.

Consider the function $y=x^{4}$.

Interchange the variables x and y.

$x=y^{4}$

Now, solve $x=y^{4}$ for $y$.

$\pm \sqrt[4]{x}=y$

Therefore, the function has no inverse.

Hence, the function $x=y$ and $y=x^{3}$ are inverse function.