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Functions and relations (domain and range) — KCSE Mathematics

KCSE Mathematics · 96 practice questions · 4 syllabus objectives · 4 revision lessons

32 easy34 medium30 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Define a relation and a function; determine the domain and range of a function from a set of ordered pairs, mapping diagram or equation

Distinguish between one-to-one, many-to-one and one-to-many mappings; determine which are functions

Find the composite function fg(x) and the inverse function f⁻¹(x) for simple functions

Functions and relations (domain and range)

Revision Notes

Concise lesson notes for Functions and relations (domain and range), written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Relations and Functions

A relation is a set of ordered pairs, where each pair consists of an input and an output. A function is a specific type of relation where each input is associated with exactly one output.

To determine the domain and range of a function:

  • Domain: The set of all possible input values (x-values).
  • Range: The set of all possible output values (y-values).

Example 1: Given the set of ordered pairs {(1, 2), (2, 3), (3, 4)}:

  • Domain: {1, 2, 3}
  • Range: {2, 3, 4}

Example 2: For the function defined by the equation y = 2x + 1:

  • Domain: All real numbers (since x can take any value).
  • Range: All real numbers (since y can also take any value).

Understanding these concepts is crucial for solving problems related to functions in mathematics.

Key points to remember

  • A relation is a set of ordered pairs.
  • A function has one output for each input.
  • Domain is the set of all input values.
  • Range is the set of all output values.
  • Use ordered pairs or equations to find domain and range.

Worked example

Determine the domain and range for the relation {(1, 5), (2, 6), (3, 7)}.

  • Domain: {1, 2, 3}
  • Range: {5, 6, 7}

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More lessons in this topic

Lesson 2: Understanding Mappings in Functions

Objective: Distinguish between one-to-one, many-to-one and one-to-many mappings; determine which are functions

In mathematics, mappings describe relationships between sets. There are three types of mappings: one-to-one, many-to-one, and one-to-many.

  • One-to-One Mapping: Each element in the domain maps to a unique element in the range. For example, if f(x) = 2x, then f(1) = 2 and f(2) = 4. Here, each input has a distinct output.
  • Many-to-One Mapping: Multiple elements in the domain map to a single element in the range. For instance, if f(x) = x², then f(2) = 4 and f(-2) = 4. Both inputs give the same output.
  • One-to-Many Mapping: An element in the domain maps to multiple elements in the range. For example, if f(x) = √x, then f(4) = 2 and f(4) = -2. This is not a function because one input has multiple outputs.

To determine if a mapping is a function, check if each input has exactly one output. If it does, it’s a function; if not, it’s not a function.

  • One-to-one: unique outputs for each input.
  • Many-to-one: multiple inputs have the same output.
  • One-to-many: one input leads to multiple outputs, not a function.
  • A function has exactly one output for each input.

Determine if the mapping f: {1 -> 2, 2 -> 3, 3 -> 2} is a function. Answer: Yes, it is a function because each input has one output.

Lesson 3: Composite and Inverse Functions

Objective: Find the composite function fg(x) and the inverse function f⁻¹(x) for simple functions

In mathematics, understanding composite functions and inverse functions is essential. Composite functions are formed by combining two functions. If we have two functions, f(x) and g(x), the composite function fg(x) is defined as fg(x) = f(g(x)).

To find the inverse function f⁻¹(x), we switch the roles of x and y in the equation y = f(x) and then solve for y. This process effectively 'reverses' the function.

Example 1: Finding Composite Function Let f(x) = 2x + 3 and g(x) = x².

  • To find fg(x):
    • Substitute g(x) into f:
    • fg(x) = f(g(x)) = f(x²) = 2(x²) + 3 = 2x² + 3.

Example 2: Finding Inverse Function Let f(x) = 3x - 5.

  • To find f⁻¹(x):
    • Replace f(x) with y: y = 3x - 5.
    • Switch x and y: x = 3y - 5.
    • Solve for y: y = (x + 5)/3.
    • Thus, f⁻¹(x) = (x + 5)/3.
  • Composite function fg(x) combines two functions f and g.
  • To find fg(x), substitute g(x) into f.
  • Inverse function f⁻¹(x) switches x and y in f(x).
  • Solve for y to determine the inverse function.
  • Understanding these concepts is crucial for function analysis.

Find fg(x) for f(x) = x + 1 and g(x) = 2x.

  • fg(x) = f(g(x)) = f(2x) = 2x + 1.
Lesson 4: Understanding Domain and Range of Functions

Objective: Functions and relations (domain and range)

In mathematics, the domain of a function is the set of all possible input values (x-values) that the function can accept. The range is the set of all possible output values (y-values) that the function can produce. To identify the domain and range:

  • Domain: Look for restrictions in the function, such as division by zero or square roots of negative numbers.
  • Range: Determine the possible values of y by analyzing the function's behavior.

For example, consider the function f(x) = 1/(x-2):

  • Domain: x cannot be 2 (as it makes the denominator zero), so the domain is all real numbers except x = 2.
  • Range: Since f(x) can take any real number except zero, the range is all real numbers except y = 0.

When working with functions, always state the domain and range clearly to score full marks in your KCSE exams.

  • Domain includes all valid x-values for the function.
  • Range includes all possible y-values produced by the function.
  • Identify restrictions like division by zero or square roots.
  • Use interval notation for clarity in answers.
  • Graphing can help visualize domain and range.

Find the domain and range of f(x) = √(x-1).

  • Domain: x ≥ 1 (x must be 1 or greater).
  • Range: y ≥ 0 (output values start from 0).

Sample Questions

Read 3 questions and answers free. Sign up to access all 96 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER3 marks

Let the function f(x) = x² + 2 for x ≥ 0. (a) Find f(4). (b) State the range of f(x). (3 marks)

Answer & marking scheme

Part (a) — 1 mark
f(4) = 4² + 2 = 16 + 2 = 18 (1 mk)
Part (b) — 2 marks
Range is [2, ∞) since f(x) = x² + 2 for x ≥ 0 (1 mk)
Minimum value occurs at x = 0, f(0) = 2 (1 mk)
2
easySHORT ANSWER2 marks

Identify the type of mapping represented by the function f(x) = 2x + 3. Explain your reasoning. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
f(x) is a one-to-one function because each input x produces a unique output f(x) (1 mk)
No two different x values yield the same f(x), satisfying the one-to-one condition (1 mk)
3
easySHORT ANSWER3 marks

Identify whether the relation defined by the set of pairs R = {(1, 2), (2, 3), (3, 4), (1, 5)} is a function. Justify your answer. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
R is not a function because the input 1 is mapped to two different outputs (2 and 5) (1 mk)
A function must have a unique output for each input (1 mk)
Thus, the mapping fails the definition of a function (1 mk)
4

A function f is defined by the set of ordered pairs f = {(1, 2), (2, 3), (3, 4), (4, 5)}. (a) State the domain of f. (b) Determine the range of f. (2 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Functions and relations (domain and range)" cover?

Functions and relations (domain and range) covers Define a relation and a function; determine the domain and range of a function from a set of ordered pairs, mapping diagram or equation; Distinguish between one-to-one, many-to-one and one-to-many mappings; determine which are functions; Find the composite function fg(x) and the inverse function f⁻¹(x) for simple functions, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Functions and relations (domain and range)?

HighMarks has 96 Functions and relations (domain and range) practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Functions and relations (domain and range) for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

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