Question

homework assignment 1.7: inverse functions
score: 11.5/15 answered: 12/15
question 13
given: $f(x) = \frac{1x - 4}{6x - 2}$
find the inverse function, $f^{-1}(x)$.
$f^{-1}(x) = $
question help: video written example

Explanation:

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

We start by letting \( y = f(x) \), so we have the equation:
\[
y = \frac{x - 4}{6x - 2}
\]

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

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

Step3: Solve for \( y \)

First, multiply both sides of the equation by \( 6y - 2 \) to eliminate the denominator:
\[
x(6y - 2) = y - 4
\]
Expand the left - hand side:
\[
6xy-2x=y - 4
\]
Now, get all the terms with \( y \) on one side and the other terms on the opposite side. Subtract \( y \) from both sides and add \( 2x \) to both sides:
\[
6xy - y=2x - 4
\]
Factor out \( y \) from the left - hand side:
\[
y(6x - 1)=2x - 4
\]
Finally, divide both sides by \( 6x - 1 \) to solve for \( y \):
\[
y=\frac{2x - 4}{6x - 1}
\]
Since \( y = f^{-1}(x) \), we have found the inverse function.

Answer:

\( f^{-1}(x)=\frac{2x - 4}{6x - 1} \)