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

The function
`x=y` and
`y=x^(3)` are inverse function.

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 √(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.

Answer 2

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.