To find the inverse of the given function [tex]\( f(x) = 4x - 12 \)[/tex], follow these detailed steps:
1. Start with the given function:
[tex]\[ y = 4x - 12 \][/tex]
2. Swap the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Rewrite the equation to reflect this swap:
[tex]\[ x = 4y - 12 \][/tex]
3. Solve for [tex]\(y\)[/tex] to find the inverse function:
- First, add 12 to both sides of the equation:
[tex]\[ x + 12 = 4y \][/tex]
- Next, divide both sides of the equation by 4:
[tex]\[ y = \frac{x + 12}{4} \][/tex]
4. Express the result in the standard inverse function notation [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x + 12}{4} \][/tex]
5. Identify the slope and intercept of the inverse function:
- The slope (coefficient of [tex]\(x\)[/tex]) in [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ \frac{1}{4} \][/tex]
- The y-intercept is:
[tex]\[ \frac{12}{4} = 3 \][/tex]
Thus, the inverse function can be written as:
[tex]\[ f^{-1}(x) = \frac{1}{4}x + 3 \][/tex]
Therefore, the values for [tex]\( f^{-1}(x) \)[/tex] are:
[tex]\[
f^{-1}(x) = 0.25x + 3.0
\][/tex]
So, the inverse of the given function is:
[tex]\[
f^{-1}(x) = 0.25x + 3.0
\][/tex]
