Functions, composition and inverses
Track the order of operations and the allowed inputs of a function.
Before you begin
- Algebraic substitution
- Linear equations
Compose functions in the stated order
A function assigns exactly one output to each permitted input. In f(g(x)), first calculate g(x), then use its output as the input to f. Order generally matters: f(g(x)) and g(f(x)) need not agree. The inner output must also lie in the outer function's domain for the composition to be defined.
Undo a one-to-one function
An inverse reverses a function's input-output relationship. To find a simple inverse, write y = f(x), rearrange for x, then rename the input variable. An inverse function exists on a chosen domain only when outputs identify inputs uniquely. For x², restricting x to nonnegative values gives the inverse √x; without restriction, +2 and −2 both map to 4.
Worked example
For f(x) = 2x + 3 and g(x) = x², find f(g(2)).
Show the worked solution
- Apply the inner function: g(2) = 4.
- Use 4 as the input to f: f(4) = 2 × 4 + 3.
- Do not reverse the order; g(f(2)) would be 49.
11.
Try it yourself
What is the inverse of f(x) = 3x − 6 on the real numbers?
Common mistakes
- f⁻¹(x) does not mean 1/f(x).
What can you explain now?
If f(x) = 5x + 1, find f⁻¹(16) and verify it.
Compare with the explanation
3.
The inverse is (x − 1)/5, so (16 − 1)/5 = 3. Substituting 3 in the original gives 16.
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Common Core mathematics standards ↗