Question

Which function has an inverse function?

Answer 1

Answer: Option B

Step-by-step explanation:

By definition, only those functions that are one to one have an inverse function.

A function is one by one if there are not two different input values, $x_1$ and $x_2$, that have the same output value y

Note that the function $f(x)= \frac{|x+3|}{5}$  is not a one-to-one function

When x=2  $f(x)= \frac{|2+3|}{5}=1\ ,\ \ y=1$

When x=8  $f(x)= \frac{|-8+3|}{5}=1\ \ ,\ y=1$

Note that the function $f(x)= \frac{x^4}{7}+ 27$  is not a one-to-one function

When x=1 $f(x)= \frac{(1)^4}{7}+27\ ,\ \ y=\frac{190}{7}$

When x=-1  $f(x)= \frac{(-1)^4}{7}+27\ ,\ \ y=\frac{190}{7}$

Note that the function $f(x)= \frac{1}{x^2}$  is not a one-to-one function

When x=1 $f(x)= \frac{1}{(1)^2}\ ,\ \ y=1$

When x=-1  $f(x)= \frac{1}{(-1)^2}\ ,\ \ y=1$

Then the answer is the option B.

You can verify that The function $f (x) = x ^ 5-3$ is a one-to-one function and therefore its inverse is a function