Functions

Objectives

After completing this section, you should be able to:

  1. Define Relation, Domain, and Range
  2. Identify Functions
  3. Use the Vertical Line Test for Functions
  4. Function Notation
  5. Identify Linear Functions
  6. Function Values from a Graph
  7. Increasing, Decreasing, and Constant

Define Relation, Domain, and Range

A relation describes a correspondence between two or more variables. Each of the following is an example of a relation.

  • x + 2y = 10
  • x2 + y2 = 25
  • {(x, y) | x > y }
  • {(4, 1), (2, -3), (0, 4), (6, 1)}

The domain of the relation is the set of all possible first coordinates. When we graph, we think of the first coordinate in terms of x.

The range of the relation is the set of all possible second coordinates. When we graph, we think of the second coordinate in terms of y.

Example 1.

Find the domain and range of the relation: {(7, 4), (-3, -3), (1, -2), (4, -6)}.

Solution

The domain consists of the set of all of the first coordinates: {7, -3, 1, 4}.

The range consists of the set of all of the second coordinates: {4, -3, -2, -6}.

Identify Functions

A function is defined as a relation in which each member of the domain is matched to exactly one member of the range. In other words, no two ordered pairs can have the same first coordinate and different second coordinate.

The relation shown in Example 1 is an example of a function since no ordered pairs have the same first coordinate.
Example 2.

Determine which of the following relations represents a function. If the relation does represent a function, then give the domain of the function.

a) {(10, 6), (23, -1), (38, 6), (10, 4), (59, 4)}

b) y = √ x + 5

c) x = y2 − 3

d) y = 4/(x - 2)

Solution

a) The set {(10, 6), (23, -1), (38, 6), (10, 4), (59, 4)} does not represent a function since the first coordinate, 10, corresponds to both 6 and 4.
 
b)

y = √ x + 5 does represent a function since every x-value that you substitute into the equation will produce a different y-value.

In order to determine the domain of the function, we need to consider the square root. The domain of a function is the set of all real numbers that are meaningful replacements for x. To find the domain, first decide if there are any values of x that are not meaningful replacements.

In this course, the types of expressions that we will consider which have values for x that are not meaningful are fractions, since they are undefined when the denominator is equal to zero, and square roots, since the square root of a negative number is imaginary.

In this case, we need to determine for which values for x, the expression √ x + 5 is real.

To do this, set the expression under the radical sign (the radicand) to be greater than or equal to zero, and solve for x.

The domain will be the solution set to x + 5 > 0. Therefore, the domain of y = √ x + 5 is x > −5. In interval notation, the domain is given as [−5, ∞).
 
c)

x = y2 − 3 does not represent a function since it is possible to find an x-value that corresponds to two different y-values.

If you select x = 1, and substitute it into the equation, then the equation becomes 1 = y2 − 3, which has two solutions. Both y = 2 and y = −2 are solutions.

1 = (−2)2 − 3 and 1 = 22 − 3
 
d) does represent a function. Every x-value that you substitute into the equation will produce a different y-value.

In order to determine the domain of the function, we need to consider the fraction. The domain of the function will represent all meaningful values of x that can be substituted into this equation.

In this case, x cannot equal 2 since the number 2 will make the denominator equal to zero and the fraction will be undefined.

Use the Vertical Line Test for Functions

If we have the graph of a relation, we can determine if the relation represents a function by using a vertical line test.

Vertical Line Test.

If a vertical line can be drawn anywhere on the graph of a function so that the vertical line crosses more than one point, then the graph does not represent a function.

Example 3.

Use the vertical line test to determine which of the following relations represents a function.

a) graph a b) graph b c) graph c

Solution

a) This is not a function since a vertical line can be drawn so that it crosses more than one point. vertical line test on graph a
b) This is a function since any vertical line that can be drawn will only cross one point. vertical line test on graph b
c) This is a function since any vertical line that can be drawn will only cross one point.

Function Notation

Function notation, such as y = f(x), illustrates the input and output process of a function.

f(x) is read “f of x,” where f is the name of a function and x is the value that is used as input into the function.

The value of a function is the output of the function, or the y -value that corresponds to an input value, x .

If f (x) = 5x − 7, then f (4) = 5(4) − 7 = 20 − 7 = 13. f (4) = 13.

To calculate the value of a function for any given x -value, just input a number into the function and simplify.

Example 4.

Given f (x) = x2 − 3x + 2, find

  a) f (0) b) f (2) c) f (t ) d) f (3k ) e) f (x + 9)

Solution

a)
f (0) = (0)2− 3(0) + 2 = 0 − 0 + 2 = 2
 f (0) = 2
b) f (2) = (2)2− 3(2) + 2 = 4 − 6 + 2 = 0 f (2) = 0
c) f ( t ) = ( t )2− 3( t ) + 2 = t2− 3 t + 2 f ( t ) = t2− 3 t + 2
d) f (3 k ) = (3 k )2− 3(3 k ) + 2 = 9 k2− 9 k + 2 f (3 k ) = 9 k2− 9 k + 2
e) f ( x + 9) = ( x + 9)2− 3( x + 9) + 2

= ( x2 + 18 x + 81) − 3 x − 27 + 2

= x2 + 15 x + 56

 f ( x + 9) = x2 + 15 x + 56
 

Identify Linear Functions

A linear function is defined as f(x) = mx + b, for real numbers m and b.

Recall that the notation for a linear function matches that of the slope-intercept form of an equation of a straight line: y = mx +b. In either case, m represents the slope of the line and (0, b) represents the y-intercept.

Example 5.

Graph the function f(x) = 4x − 1.

Solution
  f(x) = 4x − 1 is a linear function. m = 4 and the y-intercept is the point (0, −1). line through (0, -1) with slope 4

Example 6.

The linear function, f(x) = 114x + 29,350, provides a model for the number of post offices in a country from 1988 to 1995, where x = 0 corresponds to 1988, x = 1 corresponds to 1989, and so on. Use this model to give the approximate number of post offices in the year 1991.

Solution

We need to determine the value of x that corresponds to the year 1991.

Remember that x = 0 corresponds to the year 1988, x = 1 corresponds to 1989, and so on.

x represents the number of years from 1988 to 1991, or x = 1991 − 1988 = 3.

x = 3

Now, we want to find f(3)

f(3) = 114(3) + 29,350

= 342 + 29,350

= 29,692

In 1991, there were approximately 29,692 post offices.

Function Values from a Graph

The output of a function is the y-value of any point plotted on a graph. The input to the function is the x-value.

If a graph contains the point (6, 9), then this can be written in function form as f(6) = 9.

Example 7.

Refer to the graph shown below. Find the value of

  a) f (0) b) f (2) c) f ( 4 ) d) f (−2 ) e) f (3)

function graph

Solution
a)
This is function notation. The input to the function, the x-value is zero.

Look on the graph and select the point where x = 0.

The value of the function is the y-value of that point.

f (0) = −1 since the graph goes through the point (0, −1)

function graph
 
b) f (2) = −1 since the graph passes through the point (2, −1)
 
c) f ( 4 ) = 3 since the graph passes through the point (4, 3)
 
d) f (−2 )= 2 since the graph passes through the point (−2, 2 )
 
e) f (3) = 0 since the graph passes through the point (3, 0)

Increasing, Decreasing, and Constant

The terms, increasing, decreasing, and constant, are describing the slope of the function as you read the graph from the left to the right.

A graph is increasing whenever its slope is positive. A positive slope is one that goes up and to the right. up and right arrow

A graph is decreasing whenever its slope is negative. A negative slope is one that goes down and to the right. down and right

A graph is constant whenever its slope is zero, or when the graph is horizontal. right arrow

Example 8.

Refer to the graph shown below. Determine intervals of the domain for which the graph is increasing, decreasing, and constant.

function graph

Solution

You should read the graph from left to right.

Determine the intervals in which the graph is going up (increasing), going down (decreasing) or going straight across (constant).

When listing intervals, give only the x-values.

graph illustrating increasing and decreasing regions

The graph is increasing over the following intervals: (−∞, −2) ∪ (2, 4).

The graph is decreasing over the following intervals: (−2, −1) ∪ (4, ∞).

The graph is constant on the interval: (−1, 2).