A function is a relation where every element in the domain has exactly one corresponding element in the range. This is called the vertical line test — if a vertical line drawn through the graph touches it at more than one point, the relation is not a function.

Example: Which of these arrow diagrams represents a function?

• Diagram A: 1 → a, 2 → b, 3 → a (This IS a function — each input has one output)
• Diagram B: 1 → a, 1 → b, 2 → c (This is NOT a function — input 1 has two outputs)

All functions are relations, but not all relations are functions.

The domain is the set of all possible input values (x-values) in a relation or function. The range (or codomain) is the set of all possible output values (y-values).

Example: Consider the relation R = {(x, y): y = 2x} defined for x = {-1, 0, 1, 2, 3}.

Finding the domain: The domain is simply the set of x-values given: {-1, 0, 1, 2, 3}

Finding the range: Calculate y for each x:

• x = -1 → y = 2(-1) = -2
• x = 0 → y = 2(0) = 0
• x = 1 → y = 2(1) = 2
• x = 2 → y = 2(2) = 4
• x = 3 → y = 2(3) = 6

So the range is {-2, 0, 2, 4, 6}.

Example 2: For the function f(x) = x² with domain {-2, -1, 0, 1, 2}:

• f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2) = 4
• Domain = {-2, -1, 0, 1, 2}
• Range = {0, 1, 4}

A linear function has the form y = mx + c, where m is the gradient (slope) and c is the y-intercept. Its graph is a straight line.

Example: Graph the function f(x) = 2x - 1

Step 1: Create a table of values

Step 2: Plot the points and draw a straight line through them.

Key features:

• Gradient (m = 2): The line rises 2 units for every 1 unit it runs to the right
• y-intercept (c = -1): The line crosses the y-axis at (0, -1)

A quadratic function has the form y = ax² + bx + c, where a ≠ 0. Its graph is a parabola — a U-shaped curve.

Example: Graph the function f(x) = x² - 4x + 3

Step 1: Create a table of values

Step 2: Plot the points and draw a smooth curve through them.

Key features of quadratic graphs:

• The parabola opens upward if a > 0, downward if a < 0
• The vertex is the highest or lowest point
• The parabola is symmetrical about a vertical line called the axis of symmetry

The inverse of a relation reverses the direction of the mapping — if (a, b) is in the original relation, then (b, a) is in the inverse.

Example: Given the relation R = {(1, 2), (2, 4), (3, 6)}, find the inverse.

The inverse relation R⁻¹ = {(2, 1), (4, 2), (6, 3)}

For a function, the inverse f⁻¹ exists only if the function is one-to-one (each y-value corresponds to only one x-value).

Finding the inverse of a function:

Given f(x) = 3x + 2:

1. Write y = 3x + 2
2. Swap x and y: x = 3y + 2
3. Solve for y: x - 2 = 3y, so y = (x - 2)/3
4. Write as f⁻¹(x) = (x - 2)/3

Example: If f(x) = 5x - 10, then f⁻¹(x) = (x + 10)/5

Checking: f(2) = 0, and f⁻¹(0) = (0 + 10)/5 = 2 ✓

In everyday life, functions help us model relationships between quantities. For example, when a dalali (middleman) buys tomatoes from farmers in Mbeya and sells them at a fixed profit of 500 TZS per kilogram, the selling price can be expressed as a linear function: price = purchase price + 500. If a farmer sells at 2,000 TZS per kg, the function becomes f(x) = x + 500, giving 2,500 TZS per kg. Understanding domain and range helps the dalali know the minimum and maximum prices they can expect, while inverse functions can calculate how much to pay farmers to achieve a desired selling price.