Question

find the inverse of the one - to - one function ( f(x)=\frac{5}{6}x + 25 ).( f^{-1}(x)=square ) (use integers or fractions for any numbers in the expression.)

Explanation:

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

We start by rewriting the function \( f(x) = \frac{5}{6}x + 25 \) as \( y = \frac{5}{6}x + 25 \). This is a standard first step when finding the inverse of a function, as it helps us clearly identify the relationship between \( x \) and \( y \) that we will later reverse.

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

Next, we swap the roles of \( x \) and \( y \) to get \( x = \frac{5}{6}y + 25 \). Swapping \( x \) and \( y \) is based on the definition of an inverse function, where the input of the original function becomes the output of the inverse function and vice versa.

Step3: Solve for \( y \)

First, subtract 25 from both sides of the equation:
\( x - 25 = \frac{5}{6}y \)
Then, multiply both sides by the reciprocal of \( \frac{5}{6} \), which is \( \frac{6}{5} \), to isolate \( y \):
\( y = \frac{6}{5}(x - 25) \)
Expanding the right - hand side, we use the distributive property \( a(b - c)=ab - ac \), where \( a=\frac{6}{5} \), \( b = x \) and \( c = 25 \):
\( y=\frac{6}{5}x-\frac{6}{5}\times25 \)
\( \frac{6}{5}\times25 = 6\times5=30 \), so \( y=\frac{6}{5}x - 30 \)

Step4: Replace \( y \) with \( f^{-1}(x) \)

Now that we have solved for \( y \) in terms of \( x \), we replace \( y \) with the inverse function notation \( f^{-1}(x) \). So \( f^{-1}(x)=\frac{6}{5}x - 30 \)

Answer:

\( f^{-1}(x)=\frac{6}{5}x - 30 \)