fg(x) means f(g(x))
In fg(x), g is used first, then f is applied to the result: work from the bracket outwards.
- f(x) = 2x + 1 and g(x) = x²
- fg(3) = f(g(3)) = f(9) = 19
- gf(3) = g(f(3)) = g(7) = 49
fg and gf are usually different. As expressions, fg(x) = f(x²) = 2x² + 1, and gf(x) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1.
The inverse undoes the function
f⁻¹(x) reverses f. If f(x) = 3x − 4, f multiplies by 3 then subtracts 4, so f⁻¹ adds 4 then divides by 3.
- y = 3x − 4
- y + 4 = 3x
- x = (y + 4)/3
- f⁻¹(x) = (x + 4)/3
Check: f(5) = 11 and f⁻¹(11) = 15/3 = 5. To find f⁻¹(k), either find f⁻¹(x) first or solve f(x) = k.
f⁻¹(x) is not 1 ÷ f(x). The −1 is a label for "inverse", not a power.
Solving fg(x) = k
Write fg(x) as one expression first, then solve. With f(x) = 2x + 1 and g(x) = x − 3: fg(x) = 2(x − 3) + 1 = 2x − 5, so fg(x) = 9 gives 2x = 14 and x = 7.
See the inverse as a reflection
On the graphing calculator, plot y = 2x + 3, its inverse y = (x − 3)/2, and the line y = x. Pick a point on the first line, such as (1, 5). Find (5, 1) on the second line. What does the line y = x do to the graph of a function and its inverse?