Question

Which function has an inverse function?

A.f(x)= |x+3|/5
B. f(x)= x^5-3
C. f(x)= x^4/7+27
D. f(x)= 1/x²

Answer 1

Answer: Option B

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)= (|x+3|)/(5)` is not a one-to-one function

When x=2
`f(x)= (|2+3|)/(5)=1\ ,\ \ y=1`

When x=8
`f(x)= (|-8+3|)/(5)=1\ \ ,\ y=1`

Note that the function
`f(x)= (x^4)/(7)+ 27` is not a one-to-one function

When x=1
`f(x)= ((1)^4)/(7)+27\ ,\ \ y=(190)/(7)`

When x=-1
`f(x)= ((-1)^4)/(7)+27\ ,\ \ y=(190)/(7)`

Note that the function
`f(x)= (1)/(x^2)` is not a one-to-one function

When x=1
`f(x)= (1)/((1)^2)\ ,\ \ y=1`

When x=-1
`f(x)= (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