Question

Can somebody help me with 10-13 ? I don’t know how to do the inverse of the functions

Answer 1

Problem N 10

we have the function

$f(x)=9+\sqrt{4x-4}$

Find out the inverse

Let

y=f(x)

$y=9+4x-4$

step 1

Exchange the variables (x for y and y for x)

$x=9+\sqrt{4y-4}$

step 2

Isolate the variable y

$\begin{gathered} x=9+\sqrt{4y-4} \\ x-9=\sqrt{4y-4} \\ squared\text{ both sides} \\ (x-9)^2=4y-4 \\ 4y=(x-9)^2+4 \\ y=\frac{(x-9)^2}{4}+\frac{4}{4} \\ \\ y=\frac{(x-9)^{2}}{4}+1 \end{gathered}$

therefore

The inverse function is

$f^{-1}(x)=\frac{(x-9)^{2}}{4}+1$

Problem N 11

we have the function

$f(x)=\sqrt{6x-8}+5$

Find out the inverse

Let

y=f(x)

$y=\sqrt{6x-8}+5$

Exchange the variables

$\begin{gathered} x=\sqrt{6y-8}+5 \\ isolate\text{ the variable y} \\ x-5=\sqrt{6y-8} \\ squared\text{ on both sides} \\ (x-5)^2=6y-8 \end{gathered}$

isolate the variable y

$\begin{gathered} 6y=(x-5)^2+8 \\ y=\frac{(x-5)^2}{6}+\frac{8}{6} \\ \\ y=\frac{(x-5)^{2}}{6}+\frac{4}{3} \end{gathered}$

therefore

The inverse function is equal to

$f^{-1}(x)=\frac{(x-5)^{2}}{6}+\frac{4}{3}$