Question

Find g of x, the inverse of the function f of x equals five-sixths x plus five-ninths.



Drag numbers into the blanks to complete the expression for g of x.

Response area with 2 blank spaces
g of x equals
Blank space 1 empty
x plus
Blank space 2 empty

Answer 1

To find the inverse function \( g(x) \) of the function \( f(x) = \frac{5}{6}x + \frac{5}{9} \), we can follow these steps:

1. Replace \( f(x) \) with \( y \):
\[
y = \frac{5}{6}x + \frac{5}{9}
\]

2. Solve for \( x \) in terms of \( y \):
\[
y - \frac{5}{9} = \frac{5}{6}x
\]

To isolate \( x \), multiply both sides by \( \frac{6}{5} \):
\[
x = \frac{6}{5}\left(y - \frac{5}{9}\right)
\]

3. Simplify the equation:
\[
x = \frac{6}{5}y - \frac{6}{5} \cdot \frac{5}{9}
\]
\[
x = \frac{6}{5}y - \frac{6}{9}
\]
\[
x = \frac{6}{5}y - \frac{2}{3}
\]

4. Now rearranging gives:
\[
y = \frac{5}{6}x + \frac{5}{9} \implies x = \frac{6}{5}y - \frac{2}{3}
\]

Thus, switching \( x \) and \( y \) to find the inverse, we have:
\[
g(x) = \frac{6}{5}x - \frac{2}{3}
\]

Now, the answer puts us in the position to fill in the blanks:

- Blank space 1: \(\frac{6}{5}\)
- Blank space 2: \(- \frac{2}{3}\)

So, the final expression for \( g(x) \) is:
\[
g(x) = \left(\frac{6}{5}\right)x - \frac{2}{3}
\]

Your completed response will look like this:
\[
g(x) = \frac{6}{5}x - \frac{2}{3}
\]