Functions & Graphs

In SEE mathematics, you encountered simple equations and their graphs. In Grade 11, we formalize these ideas through functions -- one of the most important concepts in all of mathematics. Functions describe relationships between quantities and appear everywhere in science and engineering.

What is a Function?

A function f from set A to set B is a rule that assigns to each element in A exactly one element in B. We write f: A --> B or y = f(x).

• Domain: The set of all valid inputs (x-values)
• Range: The set of all possible outputs (y-values)
• Codomain: The set B (which may be larger than the range)

Types of Functions

• One-to-one (Injective): Different inputs give different outputs. If f(a) = f(b), then a = b.
• Onto (Surjective): Every element in the codomain is mapped to by some input.
• Bijective: Both one-to-one and onto. These functions have inverses.

Common function types:

• Linear: f(x) = mx + c (straight line)
• Quadratic: f(x) = ax² + bx + c (parabola)
• Polynomial: f(x) = anxⁿ + ... + a1x + a0
• Absolute value: f(x) = |x| (V-shape)
• Rational: f(x) = p(x)/q(x) where q(x) is not zero

Domain and Range

Worked Example: Find the domain and range of f(x) = sqrt(x - 3).

Solution:

• Domain: x - 3 >= 0, so x >= 3, i.e., [3, infinity)
• Range: sqrt always gives non-negative values, so y >= 0, i.e., [0, infinity)

Worked Example: Find the domain of f(x) = 1/(x - 2).

Solution:

• x - 2 cannot be 0, so x cannot equal 2
• Domain: All real numbers except 2, i.e., R - {2}

Graph Transformations

Starting from the graph of y = f(x):

| Transformation | New Equation | Effect | |---------------|-------------|--------| | Vertical shift up by k | y = f(x) + k | Graph moves up | | Vertical shift down by k | y = f(x) - k | Graph moves down | | Horizontal shift right by h | y = f(x - h) | Graph moves right | | Horizontal shift left by h | y = f(x + h) | Graph moves left | | Vertical stretch by a | y = a f(x) | Stretched vertically | | Reflection in x-axis | y = -f(x) | Flipped upside down |

Composite Functions

If f and g are functions, the composite function (f o g)(x) = f(g(x)).

Worked Example: If f(x) = 2x + 1 and g(x) = x², find (f o g)(3).

Solution:

• First find g(3) = 3² = 9
• Then f(9) = 2(9) + 1 = 19
• (f o g)(3) = 19

Note: f o g is generally not the same as g o f.

Key Takeaways

• A function assigns exactly one output to each input
• Domain is the set of valid inputs; range is the set of outputs
• Graph transformations allow you to sketch new functions from known ones
• Composite functions: apply one function, then another