CSEC/CXC Functions, Relations and Graphs

Functions are special relations where each input has exactly one output. Relations describe how elements from different sets are connected, and graphs provide visual representations of these relationships.

1. Relations and Functions

A relation is any set of ordered pairs. A function is a relation where each x-value (input) corresponds to exactly one y-value (output).

Example 1: Determine if these relations are functions:

a) {(1,2), (2,4), (3,6)}
b) {(1,1), (1,2), (2,3)}

Solution:

1 2 3 2 4 6 Function

Example 2: Given f(x) = 2x² - 3x + 1, find f(2)

Solution:

Example 3: If g(x) = √(x+4), find the domain

Solution:

2. Types of Functions

2.1 Linear Functions

Example 1: Graph f(x) = -½x + 3

Solution:

(0, 3) -1 2 x y

2.2 Quadratic Functions

2.2 Quadratic Functions

Example 1: Graph f(x) = x² - 4

Solution:

(0, -4) (-2, 0) (2, 0) x y

2.3 Exponential Functions

Example 1: Graph f(x) = 2ˣ

Solution:

x y (0,1) (1,2) (2,4) y = 0 (asymptote)

3. Composite and Inverse Functions

Example 1: If f(x) = 2x + 1 and g(x) = x², find (f∘g)(x) and (g∘f)(x)

Solution:

Example 2: Find the inverse of f(x) = 3x - 2

Solution:

Glossary

Self-Assessment Questions

Question 1

Which of these represents a function?

a) {(1,2), (2,3), (1,4)}
b) {(1,1), (2,2), (3,3)}

Answer: b) is a function (each x-value has only one y-value)

Question 2

Find the domain of f(x) = 1/(x-3)

Domain: {x | x ≠ 3} or (-∞,3) ∪ (3,∞)

Question 3

If f(x) = 2x - 5 and g(x) = x² + 1, find (f∘g)(2)

Solution:
g(2) = 2² + 1 = 5
f(g(2)) = f(5) = 2(5) - 5 = 5

Question 4

Find the inverse of the function h(x) = (x + 4)/3

Solution:
y = (x + 4)/3
Swap x and y: x = (y + 4)/3
Solve for y: y = 3x - 4
h⁻¹(x) = 3x - 4

Question 5

Which graph passes the vertical line test?

Graph A Vertical line Graph B Vertical line

Answer: Graph A passes the vertical line test because no vertical line intersects the curve more than once.

Graph B fails because vertical lines like x=180 intersect the circle twice.

Question 6

Given f(x) = 3ˣ, find f(0) and f(-2)

Solution:
f(0) = 3⁰ = 1
f(-2) = 3⁻² = 1/9

Question 7

Find the range of the function f(x) = x² + 2

Solution:
x² ≥ 0 for all real x
x² + 2 ≥ 2
Range: [2, ∞)

Question 8

Determine if the relation is one-to-one: {(1,5), (2,6), (3,5)}

Answer: No (both 1 and 3 map to 5)

Question 9

Sketch the graph of y = |x - 2| and state its vertex

Solution:
V-shaped graph with vertex at (2,0)

Question 10

If f(x) = 2x + 3 and g(x) = 5 - x, find f(g(1))

Solution:
g(1) = 5 - 1 = 4
f(g(1)) = f(4) = 2(4) + 3 = 11

Question 11

Find the x-intercept(s) of the function f(x) = x² - 7x + 12

Solution:
Set f(x) = 0: x² - 7x + 12 = 0
Factor: (x - 3)(x - 4) = 0
x-intercepts at x = 3 and x = 4

Question 12

A function is defined by f:x → 4x - 1. Find the value of x for which f(x) = 15

Solution:
4x - 1 = 15
4x = 16
x = 4