Further Mathematics · Lesson 1

Functions and inverses

WAECNECO

What the syllabus asks

WAEC: Pure Mathematics: Sets and functions: relations and mappings; types of functions (one-to-one, many-to-one, onto, inverse); domain, co-domain and range; composite functions; inverse functions; even and odd functions; modulus function; quadratic, rational, exponential and logarithmic functions and their graphs

NECO: Pure Mathematics: Sets and functions: relations and mappings; types of functions (one-to-one, many-to-one, onto, inverse); domain, co-domain and range; composite functions; inverse functions; even and odd functions; modulus function; quadratic, rational, exponential and logarithmic functions and their graphs

What you will learn

Distinguish relations and functions; classify mappings.
Find domains and ranges, including restricted domains.
Evaluate and form composite functions.
Find and verify inverse functions, including rational functions.
Relate graphs of f and f⁻¹; identify even and odd functions.
Work with quadratic, modulus, exponential and logarithmic functions.

Key points

A function gives each input exactly one output; many-to-one is allowed, one-to-many is not.
Domain excludes zero denominators and negative square roots; range is the set of outputs.
fg(x) = f(g(x)): apply g first; fg ≠ gf in general.
f⁻¹: y = f(x), make x the subject, swap; check f(f⁻¹(x)) = x.
Only one-to-one functions have inverses; restrict the domain if needed.
Graph of f⁻¹ = reflection of f in y = x; domain of f⁻¹ = range of f.
Even: f(−x) = f(x); odd: f(−x) = −f(x).
Quadratic a(x − h)² + k: vertex (h, k); range from k.
|x| < a ⇔ −a < x < a; |x| > a ⇔ x < −a or x > a.
aˣ and logₐ x are inverse functions.

Start here: Relations, mappings and functions

A relation links elements of one set to elements of another. A function (or mapping) is a special relation: every input has exactly one output.

Think of a school register: each student has exactly one admission number. "Student → admission number" is a function. But "class → student" is not a function, because one class contains many students.

Types of mapping

One-to-one: different inputs give different outputs. Example: x → 2x + 1.
Many-to-one: different inputs can share an output. Example: x → x² (both 3 and −3 map to 9). Still a function.
One-to-many: one input gives several outputs. Example: x → ±√x. Not a function.

The vertical line test: a graph represents a function if no vertical line crosses it more than once. A circle fails the test; a parabola y = x² passes it.

Worked example 1.1

A = {1, 2, 3}, B = {4, 5, 6, 7}. Which relations are functions from A to B? (a) {(1, 4), (2, 5), (3, 6)} (b) {(1, 4), (1, 5), (2, 6), (3, 7)} (c) {(1, 5), (2, 5), (3, 5)}.

(a) Each of 1, 2, 3 has one image: function, one-to-one.
(b) 1 has two images (4 and 5): not a function. We check each input separately.
(c) Each input has one image (all 5): function, many-to-one.

(a) and (c) are functions

⚠️ Common mistake: thinking many-to-one relations are not functions. Many inputs can share an output; what is forbidden is one input with many outputs.

Try it yourself. Is "person → mother" a function? Is "mother → child" a function?

Show solution

Person → mother: yes (each person has one biological mother; many-to-one). Mother → child: no (a mother may have several children).

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Practice questions

1. f(x) = x² − 3x. Find f(−2) and f(a − 1).

Show worked answer

10; a² − 5a + 4.

2. State the domain of f(x) = √(2x + 8).

Show worked answer

x ≥ −4.

3. f(x) = 3x − 2, g(x) = x². Find fg(x) and gf(−1).

Show worked answer

3x² − 2; gf(−1) = (−5)² = 25.

4. Find f⁻¹(x) for f(x) = (2x − 1)/(x + 3), x ≠ −3.

Show worked answer

f⁻¹(x) = (3x + 1)/(2 − x), x ≠ 2.

5. Is f(x) = x⁵ + x even, odd or neither?

Show worked answer

Odd.

Key words

Relation: Any set of ordered pairs linking two sets.
Function (mapping): A relation in which each input has exactly one output.
Domain: The set of inputs.
Co-domain: The set into which outputs fall.
Range (image set): The set of outputs actually produced.
Image: The output f(a) of an input a.
One-to-one (injective): Different inputs always give different outputs.
Many-to-one: Two or more inputs share an output.
Onto (surjective): Every element of the co-domain is an output.
Bijective: Both one-to-one and onto.
Composite function: A function of a function, fg(x) = f(g(x)).
Inverse function: f⁻¹, which reverses f.

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