Question

Find the inverse of the function below and write it in the formyequals=f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x).​(b) Verify the relationshipsf(f^-1(x)) and f^-1(f(x))=x​f(x)=3x+5(a) f^-1(x)=.....

Answer 1

The inverse of the function f(x) = 3x + 5 is
`f^(-1)(x) = (x - 5)/(3)`

How to determine the inverse of the function

From the question, we have the following parameters that can be used in our computation:

f(x) = 3x + 5

Express the function as an equation

So, we have

y = 3x + 5

Swap the occurrence of x and y in the equation

This gives

x = 3y + 5

Subtract 5 from all sides

3y = x - 5

So, we have

`y = (x - 5)/(3)`

Express as an inverse function

`f^(-1)(x) = (x - 5)/(3)`

`f^(-1)(x) = (x - 5)/(3)`

Verifying the relationship
`f^(-1)(f(x))` and
`f(f^(-1)(x))`

We have

`f^(-1)(f(x)) = (3x + 5 - 5)/(3)`

`f^(-1)(f(x)) = (3x )/(3)`

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

Also, we have

`f(f^(-1)(x)) = 3 * (x - 5)/(3) + 5`

`f(f^(-1)(x)) = x - 5 + 5`

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

Hence, the inverse of the function is
`f^(-1)(x) = (x - 5)/(3)`

Question

Find the inverse of the function below and write it in the form y = f^-1(x)

Verify the relationshipsf(f^-1(x)) and f^-1(f(x))=x​

f(x)=3x+5

Answer 2

a.
`f^(-1)(x)=(x-5)/(3)`.

b. See below

Explanation:

The given function is:
`f(x)=3x+5`

To find the inverse function, we let
`y=3x+5`.

We interchange x and y to obtain:
`x=3y+5`.

We now solve for y;

First add -5 to both sides of the equation;

`x-5=3y`.

Divide both sides by 3

`(x-5)/(3)=y`.

Or

`y=(x-5)/(3)`.

The inverse function is
`f^(-1)(x)=(x-5)/(3)`.

b.

Let us now verify that;

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

`\implies f^(-1)(f(x))=(3x+5-5)/(3)`

`\implies f^(-1)(f(x))=(3x)/(3)`

`\implies f^(-1)(f(x))=(x)/(1)`

`\implies f^(-1)(f(x))=x`

`\boxed{Q.E.D}`