If $f$ is one-to-one, find an equation for its inverse.
\[
f(x)=8 x+7
\]
A. $f^{-1}(x)=\frac{x+7}{8}$
B. $f^{-1}(x)=\frac{x-7}{8}$
C. $f^{-1}(x)=\frac{x}{8}-7$
D. The function is not one-to-one.
Answer
So, the inverse function is $f^{-1}(x) = \frac{x - 7}{8}$, which corresponds to option B.
Step 1 ：Since $f$ is one-to-one, it has an inverse. We can find the inverse by swapping $x$ and $y$ and solving for $y$.
Step 2 ：First, we write the function $f(x)$ as $y = 8x + 7$.
Step 3 ：Next, we swap $x$ and $y$ to get $x = 8y + 7$.
Step 4 ：Now, we solve for $y$ to find the inverse function. Subtract 7 from both sides to get $x - 7 = 8y$.
Step 5 ：Finally, divide both sides by 8 to isolate $y$ and get $y = \frac{x - 7}{8}$.
Step 6 ：So, the inverse function is $f^{-1}(x) = \frac{x - 7}{8}$, which corresponds to option B.
