Question

5. Find the inverses of the following functions:
a. f(x) = (2 / 3 x) − 10
b. g(x) = 2(x + 4)3
c. h(x) = 1 / x−2 , xx ≠ 2

Answer 1

Answer:

a)  

Step 1. We’ll first replace f(x) with y

$y=\frac{2x}{3}-10$

Step 2. Replace all x’s with y and all y’s with x.

$x=\frac{2y}{3} -10$

Step 3. Solve for y. Then, replace y with f−1(x)

$x+10=\frac{2y}{3}\\\frac{ 3(x+10)}{2}=y\\ \frac{3x+30}{2}=y=f^{-1}(x)$

Then, $f^{-1}(x)=\frac{3x+30}{2}$

b)

Step 1. $y=2(x+4)3=6(x+4)=6x+24$

Step 2. $x=6y+24$

Step 3.

$x-24=6y\\\frac{x-24}{6}=y=g^{-1}(x)$

Then, $g^{-1}(x)=\frac{x-24}{6}$

c)

Step 1. $y=\frac{1}{x-2}$

Step 2. $x=\frac{1}{y-2}$

Step 3.

$y-2=\frac{1}{x}\\y=\frac{1}{x}+2\\h^{-1}(x)=\frac{1}{x}+2$ with $x\neq 0$