1. Determine if 2 Functions are Inverses by Compositions

2. You can use composition of functions
to verify that 2 functions are inverses.
The inverse functions “undo” each other,
When you compose two inverses…
the result is the input value of x.
If f(g(x)) = g(f(x)) = x
Then f(x) and g(x) are inverse functions

3. Because f(g(x)) = g(f(x)) = x, they are inverses.
Example 1:
Determine by composition if the functions are inverses functions.
Because f(g(x)) = g(f(x)) = x, they are inverses.

4. Find the composition f(g(x)).
Example 2:
Determine by composition whether each pair of functions are inverses. 1 3
f(x) = 3x – 1 and g(x) = x + 1
Find the composition f(g(x)).
= x + 3 – 1
f(g(x)) = x + 2
The functions are NOT inverses.

5. Because f(g(x)) = g(f(x)) = x, they are inverses.
Example 3
Determine whether are inverse functions.
Because f(g(x)) = g(f(x)) = x, they are inverses.

6. Because f(g(x)) = g(f(x)) = x, they are inverses.
Example 4
Determine by composition whether each pair of functions are inverses. 3 2
f(x) = x + 6 and g(x) = x – 9
Find the composition f(g(x)) and g(f(x)).
= x –
= x + 9 – 9
Because f(g(x)) = g(f(x)) = x, they are inverses.

7. Find the compositions f(g(x)) and g(f(x)).
Example 5
f(x) = x2 + 5 and for x ≥ 0
Find the compositions f(g(x)) and g(f(x)).
Substitute for x in f.
= x 10 x -
Simplify.
Because f(g(x)) ≠ x, f (x) and g(x) are NOT inverses.