Pc An Introduction To Functions

An Introduction to Functions

A relation is a rule of correspondence that relates two sets. For instance, the formula I = 500r describes a relation between the amount of interest I earned in one year and the interest rate r. In mathematics, relations are represented by sets of ordered pairs (x, y) .

A function from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is called the domain of the function.

The set B is called the range of the function.

Function Set B Set A

y y

x y

x

y

x x

Domain

y

Range

Example: Determine whether the relation represents y as a function of x. a)

{(-2, 3), (0, 0), (2, 3), (4, -1)} Function

b)

{(-1, 1), (-1, -1), (0, 3), (2, 4)} Not a Function

Functions represented by equations are often named using a letter such as f or g.

The symbol f (x), read as “the value of f at x” or simply as “f of x”, is the element in the range of f that corresponds with the domain element x. That is, y = f (x)

The domain elements, x, can be thought of as the inputs and the range elements, f (x), can be thought of as the outputs.

Function

Input

Output

f

x

f (x)

To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function. Example: Let f (x) = x2 – 3x – 1. Find f (–2). f (x) = x2 – 3x – 1 2

f (–2) = (–2) – 3(–2) – 1

Substitute –2 for x.

f (–2) = 4 + 6 – 1

Simplify.

f (–2) = 9

The value of f at –2 is 9.

Example: Let f (x) = 4x – x2. Find f (x + 2). f (x) = 4x – x2 f (x + 2) = 4(x + 2) – (x + 2)2

Substitute x + 2 for x.

f (x + 2) = 4x + 8 – (x2 + 4x + 4)

Expand (x + 2)2.

2

f (x + 2) = 4x + 8 – x – 4x – 4

Distribute –1.

f (x + 2) = 4 – x2

The value of f at x + 2 is 4 – x2.

The domain of a function f is the set of all real numbers for which the function makes sense. Example: Find the domain of the function f (x) = 3x +5 Domain: All real numbers

Example: Find the domain of the function

f ( x) = x − 3 The function is defined only for x-values for which x – 3 ≥ 0. Solving the inequality yields x–3≥0 x≥3 Domain: {x| x ≥ 3}

Example: Find the domain of the function x +2 g ( x) = 2 x −1

The x values for which the function is undefined are excluded from the domain. The function is 2 undefined when x – 1 = 0. x2 – 1 = 0 (x + 1)(x – 1) = 0 x=±1 Domain: {x| x ≠ ± 1}