Section 5.2 Functions

Relation, Domain, and Range Definition A relation is a set of ordered pairs. The domain of a relation is the set of all values of the independent variable, and the range of the relation is the set of all values of the dependent variable. We can think of a relation as a machine in which values of x are “inputs” and values of y are “outputs.” In general, each member of the domain is an input, and each member of the range is an output.

A relation described by a table. Relation, Domain, and Range A relation described by a table. A relation described by a graph.

Think of a relation as an input-ouput machine. Relation, Domain, and Range Think of a relation as an input-ouput machine.

Function Definition A function is a relation in which each input leads to exactly one output.

Deciding whether an Equation Describes a Function Example 1 Is the relation y = x + 2 a function? Find the domain and range of the relation. Solution Let’s consider some input-output pairs in the table on the next slide.

Each input leads to just one output-namely, the input Deciding whether an Equation Describes a Function Solution continued Each input leads to just one output-namely, the input increased by 2-so the relation y = x + 2 is a function.

Deciding whether an Equation Describes a Function Solution continued The domain of the relation y = x + 2 is the set of all real numbers, since we can add 2 to any real number. The range of y = x + 2 is also the set of real numbers, since any real number is the output of the number that is 2 units less than it.

Is the relation y = ±x a function? Deciding whether an Equation Describes a Function Example 2 Is the relation y = ±x a function? Solution If x = 1, then y = ±1. So, the input x = 1 leads to two outputs: y = −1 and y = 1. Therefore, the relation y = ±x is not a function.

Is the relation y2 = x a function? Deciding whether an Equation Describes a Function Example 3 Is the relation y2 = x a function? Solution Let’s consider the input x = 4. The input x = 4 leads to two outputs: y = −2 and y = 2. So, the relation y2 = x is not a function.

Is the relation described by the table a function? Deciding whether a Table Describes a Function Example 4 Is the relation described by the table a function? Solution The input x = 1 leads to two outputs: y = 3 and y = 5. So, the relation is not a function.

Vertical-Line Test A relation is a function if and only if every vertical line intersects the graph of the relation at no more than one point. We call this requirement the vertical-line test.

Determine whether the graph represents a function. Deciding whether a Graph Describes a Function Example 6 Determine whether the graph represents a function.

Deciding whether a Graph Describes a Function Solution 1. Since the vertical line (red) sketched intersects the circle more than once, the relation is not a function.

Deciding whether a Graph Describes a Function Solution 2. Each vertical line sketched intersects the curve at one point. In fact, any vertical line would intersect this curve at just one point. So, the relation is a function.

Finding the Domain and Range Example 9 Use the graph of the function to determine the function’s domain and range. 1. 2.

Finding the Domain and Range Solution The domain is the set of all x-coordinates of points in the graph. Since there are no breaks in the graph, and since the leftmost point is (−4, 2) and the rightmost point is (5,−3), the domain is −4≤ x ≤ 5. The range is the set of all y-coordinates of points in the graph. Since the lowest point is (5, −3) and the highest point is (2, 4), the range is −3 ≤ y ≤ 4.

Finding the Domain and Range Solution continued The graph extends to the left and right indefinitely without breaks, so every real number is an x-coordinate of some point in the graph. The domain is the set of all real numbers.

Finding the Domain and Range Solution continued The output −3 is the smallest number in the range, because (1, −3) is the lowest point in the graph. The graph also extends upward indefinitely without breaks, so every number larger than −3 is also in the range. The range is y ≥ −3.