Q.Find the inverse of the function $f(x) = 3x - 5$, and verify that $f(f^{-1}(x)) = x$.

The inverse of a linear function is found by writing $y=f(x)$ and solving algebraically for $x$ in terms of $y$.

Let $y = f(x) = 3x-5$. To find the inverse, solve this equation for $x$ in terms of $y$:

Renaming the variable, the inverse function is

Verification:

$f\big(f^{-1}(x)\big) = f\!\left(\frac{x+5}{3}\right) = 3\left(\frac{x+5}{3}\right) - 5 = (x+5) - 5 = x$ …

A frequent slip is forgetting to divide the entire right-hand side by $3$ (writing $x = y+5/3$ instead of $x=(y+5)/3$) — every term a …