Question

Hi I’m looking to get a step by step solution in solving this problem in the red

Answer 1

Given:

`\begin{gathered} f(x)=13x+2 \\ \\ g(x)=3x^2-13 \\ \\ h(x)=(13)/(x+13) \end{gathered}`

Find-:

The inverse of a function.

Explanation-:

(a)

For the inverse of a function, x change as y and y change as x and solve for 'y'

`\begin{gathered} f(x)=13x+2 \\ \\ f(y)=13y+2 \\ \\ x\rightarrow y \\ \\ y\rightarrow x \\ \\ \end{gathered}`

Then solve,

`\begin{gathered} y=13x+2 \\ \\ y-2=13x \\ \\ x=(y-2)/(13) \end{gathered}`

So, value,

`f^(-1)(y)=(y-2)/(13)`

(b)

`g(x)=3x^2-13`

So, the value is:

`g(y)=3y^2-13`

The inverse of a function is:

`\begin{gathered} x=3y^2-13 \\ \\ x\rightarrow y \\ \\ y\rightarrow x \\ \\ y=3x^2-13 \\ \\ 3x^2=y+13 \\ \\ x^2=(y+13)/(3) \\ \\ x=\sqrt{(y+13)/(3)} \end{gathered}`

So, the inverse value is:

`g^(-1)(y)=\sqrt{(y+13)/(3)}`

(c)

`h(x)=(13)/(x+13)`

Value of h(y) is:

`h(y)=(13)/(y+13)`

Then solve for inverse function,

`\begin{gathered} x=(13)/(y+13) \\ \\ x\rightarrow y \\ \\ y\rightarrow x \\ \\ y=(13)/(x+13) \\ \\ y(x+13)=13 \\ \\ x+13=(13)/(y) \\ \\ x=(13)/(y)-13 \end{gathered}`

So, inverse value is:

`h^(-1)(y)=(13)/(y)-13`