What is a function?
A function is a relation that maps every element of the domain (the set of inputs) to exactly one element of the codomain. If an input had two different outputs, it would not be a function. We write a function as f: x → 2x + 1 or f(x) = 2x + 1. The input x is the object and the output f(x) is the image. The set of all images produced is the range.
Images, objects and evaluation
To find an image, substitute the object into the rule. To find an object from a given image, form an equation and solve. Functions may be linear, quadratic, rational or absolute value. The absolute value function |x| returns the non-negative size of a number, so |−4| = 4 and |3| = 3; its range is always ≥ 0. Because |x| ignores the sign, an equation such as |2x − 5| = 3 usually has two objects: 2x − 5 = 3 gives x = 4, while 2x − 5 = −3 gives x = 1. A rational function such as (2x + 3)/(x − 1) is undefined where the denominator is zero, so x = 1 must be excluded from its domain. Recognising these restrictions is part of describing a function correctly.
Key formula / Notation
f: x → f(x), image = f(object). Absolute value: |x| = x if x ≥ 0, |x| = −x if x < 0. A relation is a function only if each object has one image (vertical line test).
Worked example
Given f(x) = 3x − 2. (a) Find the image of 5. (b) Find the object whose image is 13.
(a) f(5) = 3(5) − 2 = 15 − 2 = 13.
(b) 3x − 2 = 13 ⇒ 3x = 15 ⇒ x = 5. For g(x) = |2x − 5|, g(1) = |2 − 5| = |−3| = 3.
Remember
- Every object has exactly one image; one image may come from several objects.
- Range is the set of images actually produced, not the whole codomain.
- |x| is never negative, so an equation such as |x| = −2 has no solution.