Question

chapter 1 quiz (1.1,1.2, 1.4, 1.5, 1.7)
14 points possible answered: 13/14
question 14
given: $f(x) = \frac{6x - 6}{4x + 2}$
find the inverse function, $f^{-1}(x)$.
$f^{-1}(x) = $

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by letting \( y = f(x) \), so the function becomes \( y=\frac{6x - 6}{4x + 2} \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x=\frac{6y - 6}{4y + 2} \).

Step3: Solve for \( y \)

First, multiply both sides by \( 4y + 2 \) to get rid of the denominator:
\( x(4y + 2)=6y - 6 \)
Expand the left - hand side:
\( 4xy+2x = 6y - 6 \)
Now, collect the terms with \( y \) on one side and the other terms on the opposite side. Subtract \( 6y \) from both sides and subtract \( 2x \) from both sides:
\( 4xy-6y=-6 - 2x \)
Factor out \( y \) from the left - hand side:
\( y(4x - 6)=-6 - 2x \)
Then, solve for \( y \) by dividing both sides by \( 4x - 6 \):
\( y=\frac{-6 - 2x}{4x - 6} \)
We can factor out a - 2 from the numerator and a 2 from the denominator:
The numerator \( -6 - 2x=-2(x + 3) \) and the denominator \( 4x - 6 = 2(2x - 3) \)
So \( y=\frac{-2(x + 3)}{2(2x - 3)} \)
Cancel out the common factor of 2:
\( y=\frac{-(x + 3)}{2x - 3}=\frac{-x - 3}{2x - 3}=\frac{x + 3}{-2x + 3}=\frac{x + 3}{3 - 2x} \)

Answer:

\( f^{-1}(x)=\frac{x + 3}{3 - 2x} \)