Function notation and evaluation

The notation f(x) names the output of a function f at input x. It does not mean f multiplied by x. For f(x) = 2x² − 3x + 1, f(−2) requires brackets around −2 before squaring. For an algebraic input such as x + h, substitute the whole expression with brackets at every occurrence.

A function assigns one output to each input in its domain. Different inputs may have the same output. An equation such as f(x) = 5 asks for the input values producing output 5; evaluating f(5) asks for a single output at input 5. These two tasks can lead to entirely different numbers.

Worked example

Given f(x) = 2x² − 3x + 1, find f(−2) and solve f(x) = 1.

1. f(−2) = 2(−2)² − 3(−2) + 1 = 8 + 6 + 1.
2. For f(x) = 1, cancel the constant to obtain 2x² − 3x = 0.
3. Factor x(2x − 3) = 0.

Answer: f(−2) = 15; f(x) = 1 when x = 0 or x = 3/2