Practice Maths

L40 — Functions and Function Notation

Key Terms

Relation
A set of ordered pairs (x, y) that associates elements of one set with elements of another.
Function
A relation where each input x maps to exactly one output y — passes the vertical line test.
Function notation f(x)
Means "the output of function f when the input is x"; read as "f of x".
Vertical line test
A graph represents a function if every vertical line intersects it at most once.
Evaluating f(a)
Substituting x = a into the rule to find the corresponding output value.
Solving f(x) = k
Finding the value(s) of x that produce the output k — the reverse of evaluating.

Relations vs functions

ConceptDefinitionExample
RelationAny set of ordered pairs (x, y){(1,2), (1,3), (2,4)}
FunctionEach x-value maps to exactly one y-value{(1,2), (2,4), (3,6)}
Vertical line testGraph is a function if every vertical line crosses it at most onceParabola y=x² ✓
Many-to-oneMultiple x → same y (allowed)y = x² (both 2 and −2 give 4)
One-to-manyOne x → multiple y (NOT a function)x = y² (fails vertical line test)

Function notation

NotationMeaningExample with f(x) = 2x + 1
f(x)"f of x" — the output when input is xf(x) = 2x + 1
f(3)Substitute x = 3f(3) = 2(3)+1 = 7
f(a)Substitute x = af(a) = 2a + 1
f(x+1)Substitute x = x+1f(x+1) = 2(x+1)+1 = 2x+3
f(x) = g(x)Find x where outputs are equalSolve the equation
Domain Range 1 2 3 3 5 7 f(x) = 2x + 1
A function maps each input (domain) to exactly one output (range).
Hot Tip: A function must give exactly ONE output for each input — not zero, not two. Use the vertical line test on a graph: if any vertical line crosses the curve more than once, it's a relation but not a function.

Worked Example 1 — Is it a function?

Determine whether each relation is a function: (a) {(1,3), (2,5), (3,5), (4,7)}; (b) {(1,2), (2,4), (1,6)}; (c) y = 3x − 1.

(a) Each x-value (1, 2, 3, 4) appears exactly once. Function ✓ (many-to-one: both 2 and 3 map to 5).

(b) x = 1 appears twice (mapping to both 2 and 6). Not a function ✗.

(c) For every x, there is exactly one y. Vertical line test passes. Function ✓.

Worked Example 2 — Evaluating functions

Given f(x) = x² − 3x + 2, find: (a) f(4); (b) f(−1); (c) f(x + 1).

(a) f(4) = 4² − 3(4) + 2 = 16 − 12 + 2 = 6.

(b) f(−1) = (−1)² − 3(−1) + 2 = 1 + 3 + 2 = 6.

(c) f(x+1) = (x+1)² − 3(x+1) + 2 = x²+2x+1 − 3x − 3 + 2 = x² − x.

Worked Example 3 — Solving f(x) = k

Given g(x) = 2x² − 5, find x such that g(x) = 13.

2x² − 5 = 13 ⇒ 2x² = 18 ⇒ x² = 9 ⇒ x = ±3.

Both x = 3 and x = −3 are solutions.

See Answers ➔

  1. Relations and functions. Fluency

    Determine whether each relation is a function. Give a reason.

    • (a) {(2, 5), (3, 8), (4, 11), (5, 14)}
    • (b) {(1, 3), (2, 3), (3, 3), (4, 3)}
    • (c) {(0, 1), (1, 2), (1, 3), (2, 4)}
    • (d) {(−2, 4), (0, 0), (2, 4)}

  2. Evaluating functions. Fluency

    For f(x) = 3x − 4, find:

    • (a) f(2)
    • (b) f(0)
    • (c) f(−3)
    • (d) f(a + 1)

  3. Quadratic function evaluation. Fluency

    For g(x) = x² + 2x − 3, find:

    • (a) g(1)
    • (b) g(−2)
    • (c) g(0)
    • (d) g(x) = 5. Solve for x.

  4. Vertical line test and function notation. Fluency

    • (a) A circle with equation x² + y² = 16 — is this a function? Explain using the vertical line test.
    • (b) If h(x) = 5 for all x, what type of function is h? Evaluate h(100).
    • (c) For f(x) = 4x + 1, find x if f(x) = 17.
    • (d) For f(x) = x² and g(x) = 4x − 4, find all x where f(x) = g(x).

  5. Reading a function graph. Understanding

    The graph shows f(x) = x² − 4x + 3.

    x y −1 −2 1 2 3 1 2 −1 (1, 0) (3, 0) (2, −1) (0, 3)
    Graph of f(x) = x² − 4x + 3.
    • (a) State the x-intercepts of f(x).
    • (b) State the y-intercept of f(x).
    • (c) State the coordinates of the turning point.
    • (d) Use the graph to solve f(x) < 0. Write the solution as an inequality.

  6. Function operations. Understanding

    Given f(x) = 2x + 3 and g(x) = x² − 1:

    • (a) Find f(x) + g(x). Simplify.
    • (b) Find f(x) × g(x). Expand and simplify.
    • (c) Find f(g(2)).
    • (d) Find all values of x where f(x) = g(x).

  7. Function notation in context. Understanding

    The cost to hire a plumber is modelled by C(h) = 80h + 120, where h is the number of hours and C is the cost in dollars.

    • (a) Find C(3) and interpret its meaning.
    • (b) Find h if C(h) = 440. What does this mean?
    • (c) What does C(0) represent?
    • (d) Write an expression for C(h + 1) − C(h) and explain its meaning.

  8. Identifying functions from graphs. Understanding

    • (a) A graph is a horizontal line y = 5. Is it a function? What is f(7)?
    • (b) A graph is a vertical line x = 3. Is it a function? Explain why or why not.
    • (c) The graph of y = |x| (absolute value). Is it a function? Evaluate f(−6).
    • (d) Give an example of a real-world situation that can be modelled by a function, and explain what the inputs and outputs represent.

  9. Piecewise functions. Problem Solving

    A taxi company charges as follows: $3.50 flag fall, then $2.10 per km for the first 5 km, then $1.60 per km beyond 5 km. Let C(d) be the cost for a trip of d km.

    • (a) Write C(d) as a piecewise function.
    • (b) Find C(3), C(5), and C(8).
    • (c) Find d such that C(d) = $22.10.
    • (d) A competitor charges a flat $1.80 per km with no flag fall. For what values of d is the taxi cheaper?

  10. Function relationships. Problem Solving

    The temperature (°C) at a weather station t hours after midnight is modelled by T(t) = −0.5t² + 6t − 5 for 0 ≤ t ≤ 12.

    • (a) Find T(0) and T(12). What do these represent?
    • (b) Find the time when the temperature is at its maximum. What is the maximum temperature?
    • (c) Solve T(t) = 0 to find the times when the temperature is 0°C.
    • (d) Between which hours is T(t) > 0? Interpret this in context.