To find the inverse of the function [tex]\( f(x) = 9x + 5 \)[/tex], follow these steps:
1. Express the function using [tex]\( y \)[/tex] instead of [tex]\( f(x) \)[/tex]:
[tex]\[ y = 9x + 5 \][/tex]
2. Swap the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This is because the inverse function will swap the dependent and independent variables:
[tex]\[ x = 9y + 5 \][/tex]
3. Solve for [tex]\( y \)[/tex]. This step involves isolating [tex]\( y \)[/tex] on one side of the equation:
\begin{align}
x &= 9y + 5 \\
x - 5 &= 9y \\
y &= \frac{x - 5}{9}
\end{align}
4. Rewrite the final equation as the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x - 5}{9} \][/tex]
Now, we need to determine if this inverse relation is indeed a function. We can check this by considering if the original function [tex]\( f(x) = 9x + 5 \)[/tex] is one-to-one, meaning it passes the horizontal line test.
Since [tex]\( f(x) = 9x + 5 \)[/tex] is a linear function with a non-zero slope (specifically, a positive slope of 9), it is strictly increasing. Therefore, every horizontal line will intersect the graph of [tex]\( f(x) \)[/tex] at most once. A strictly increasing function guarantees that the inverse will also pass the vertical line test and be a function.
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x - 5}{9} \][/tex]
And indeed, [tex]\( f^{-1}(x) \)[/tex] is a function.
