Checkpoint: Inverses and a Composition of Functions Solution

Determine if the given functions are inverses.

$f (x) = - \frac{1}{2} (x - 16) + 10 g (x) = - 2 x + 36$

$[f \circ g] (x)$

$[f \circ g] (x) = - \frac{1}{2} ((- 2 x + 36) - 16) + 10 [f \circ g] (x) = - \frac{1}{2} (- 2 x + 20) + 10 [f \circ g] (x) = x - 10 + 10 [f \circ g] (x) = x$

$[g \circ f] (x)$

$[g \circ f] (x) = - 2 (- \frac{1}{2} (x - 16) + 10) + 36 [g \circ f] (x) = - 2 (- \frac{1}{2} x + 8 + 10) + 36 [g \circ f] (x) = - 2 (- \frac{1}{2} x + 18) + 36 [g \circ f] (x) = x - 36 + 36 [g \circ f] (x) = x$

Because the composition of $[f \circ g]$ and $[g \circ f]$ both equal x, the functions are inverses of one another.

Note

You could also solve by distributing $- \frac{1}{2}$ across the terms and then using substitution. Both methods result in the same answer.

Q: If the first composition of functions is not equal to x, do you need to check the next one? Explain.

A: No, because both compositions must equal x.