Section 7.6 Functions Math in Our World

Learning Objectives  Identify functions.  Write functions in function notation.  Evaluate functions.  Find the domain and range of functions.  Determine if a graph represents a function.

Relations A relation is a rule matching up two sets of objects. Relations are often represented by sets of ordered pairs. The following are examples of relations: A = {(2, 3), (– 2, 1), (12, – 3), (2, – 1)} B = {(9, 0), (– 3, 1), (3, 9), (– 1, 5), (2, 3)} C = {(x, y) | 3x + 5y = 7}

Functions A function is a relation in which each x coordinate gets paired with exactly one y coordinate. In other words, the first coordinate is never repeated with a different second coordinate. A = {(2, 3), (– 2, 1), (12, – 3), (2, – 1)} is not a function because the points (2, 3) and (2, – 1) have the same first coordinate. B = {(9, 0), (– 3, 1), (3, 9), (– 1, 5), (2, 3)} is a function because no first coordinate is repeated.

Functions A function is a relation in which each x coordinate gets paired with exactly one y coordinate. In other words, the first coordinate is never repeated with a different second coordinate. C = {(x, y) | 3x + 5y = 7} is a function because for each value of x that you substitute into the equation, there is only one possible value of y that corresponds to it.

EXAMPLE 1 Identifying Functions Defined by Equations Which of the following equations represent functions? (Assume that x represents the first coordinate.) (a) y = x 2 (b) 3x 2 + y – 2x = 5 (c) x =  y  SOLUTION (a) This is a function. Every number has only one square, so every value of x has only one associated y. (b) This is also a function. Every x will again have only one associated y. (c) This is not a function. Positive values of x will correspond to two possible values of y. (i.e. if x = 2, y can be 2 or – 2.)

Function Notation The equation y = x 2 represents a function that relates variables x and y. We call x the independent variable and y the dependent variable because its value depends on the choice of x. Another way to write the same function is f(x) = x 2. This is known as function notation, and is read aloud as “f of x equals x squared.”

Function Notation To write an equation in function notation, solve for y in terms of x, then change the letter y to the symbol f(x). Functions can also be called by names other than f. Any letter other than x or y is acceptable.

EXAMPLE 2 Writing a Function in Function Notation Write 3x – 2y = 6 in function notation. SOLUTION We need to solve the equation for y, then replace y with f(x).

Evaluating a Function When a function is written as f(x), f(2) means to find the value of the function when x = 2. This is known as evaluating a function. We call x = 2 the input, and the resulting value of the function the output.

EXAMPLE 3 Evaluating a Function Let f(x) = x 2 + 3x – 5. Find f(3), f(-2), and f(0). SOLUTION

Domain and Range The domain of a function is the set of all values of the independent variable x that result in real number values for y. The range of a function is the set of all possible y values.

EXAMPLE 4 Finding the Domain and Range of a Function Find the domain and range of each function:

EXAMPLE 4 Finding the Domain and Range of a Function SOLUTION There are no restrictions on what values x can be; therefore the domain is all real numbers. Since x 2 is never negative, the range is {y  y ≥ 0}. Since the square root of a negative number is undefined, x cannot be negative. Therefore, the domain is {x  x ≥ 0}. Since the square root of x is never negative the range is {y  y ≥ 0}.

EXAMPLE 4 Finding the Domain and Range of a Function SOLUTION Since the denominator of a fraction cannot be zero, we must exclude x = – 1. Every other x value will result in a real number output, so the domain is all real numbers except – 1, which we write as {x  x ≠ – 1}. The range is not obvious, but notice that an output of 3 would make the equation Multiplying both sides by x + 1, we get the contradiction 3x + 3 = 3x – 2, so the range is {y  y ≠ 3}.

Vertical Line Test We know that a relation is not a function if any x value corresponds to more than one output. Consider the graph of an equation shown. The two points labeled have the same x coordinate, so the equation is not a function. This indicates that if any vertical line crosses a graph more than once, the graph does not represent a function.

Vertical Line Test In this graph, there is no vertical line that crosses the graph more than once, so it is the graph of a function. Vertical Line Test for Functions If no vertical line can intersect the graph of a relation at more than one point, then the relation is a function.

EXAMPLE 5 Using the Vertical Line Test Use the vertical line test to determine whether each relation graphed represents a function.

EXAMPLE 5 Using the Vertical Line Test SOLUTION This is a function since any vertical line will cross the graph only once.

EXAMPLE 5 Using the Vertical Line Test SOLUTION This is not a function since there is a vertical line that crosses the graph in more than one place.