To check if two functions, like $f(x)$ and $g(x)$, are inverses of each other, we need to do two main things:

1. Function Composition: We have to calculate $f(g(x))$ and $g(f(x))$:

   • If $f(g(x)) = x$ for every $x$ in the range of $g$, then we meet the first requirement.
   • If $g(f(x)) = x$ for every $x$ in the range of $f$, then we meet the second requirement.

   If both of these are true, then $f(x)$ and $g(x)$ are inverses!

2. Graphing: When we look at the graphs of these functions, they are inverses if they look the same on either side of the line $y = x$.

Example:

Let’s look at the functions $f(x) = 2x + 3$ and $g(x) = \frac{x - 3}{2}$.

• Step 1: Check the compositions:

  • For $f(g(x))$: $f(g(x)) = f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x.$
  • For $g(f(x))$: $g(f(x)) = g(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x.$

• Step 2: Since both results equal $x$, we conclude that $f(x)$ and $g(x)$ are inverses of each other!