Functions and Their Graphs

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• Identify and graph linear and squaring functions.• Recognize EVEN and ODD functions

• Identify and graph cubic, square root, and reciprocal function.

• Identify and graph step and other piecewise-defined functions.

• Recognize graphs of parent functions.

OBJECTIVE :

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Linear and Squaring Functions

The graph of the linear function y= mx+b • The domain of the function is the set of all real numbers.

• The range of the function is the set of all real numbers.

• The graph has an x-intercept of (–b/m, 0) and

a y-intercept of (0, b).

• The graph is increasing if slope m 0, and decreasing if

slope m 0, and constant (horizontal line) if m = 0.

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Example 1 – Writing a Linear Function

Write the linear function for which f (1) = 3 and f (4) = 0.

Solution:

To find the equation of the line that passes through(x1, y1) = (1, 3) and (x2, y2) = (4, 0) first find the slope of the line.

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Example 1 – Solution

Next, use the point-slope form of the equation of a line.

y – y1 = m(x – x1)

y – 3 = –1(x – 1)

y = –x + 4

f (x) = –x + 4

cont’d

Point-slope form

Substitute for x1, y1 and m

Simplify.

Function notation

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Example 1 – Solution

The graph of this function is shown in Figure 1.65.

cont’d

Figure 1.65

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Linear and Squaring Functions

There are two special types of linear functions, the constant function and the identity function.

A constant function has the form

f (x) = c

and has the domain of all real numbers with a range consisting of a single real number c.

The graph of a constant function is a horizontal line, as shown inFigure 1.66. The identity function has the form f (x) = x.

Figure 1.66

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Linear and Squaring Functions

Its domain and range are the set of all real numbers. The identity function has a slope of m = 1 and a y-intercept at (0, 0).

The graph of the identity function is a line for which eachx-coordinate equals the corresponding y-coordinate. The graph is always increasing, as shown in Figure 1.67.

Figure 1.67

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Linear and Squaring Functions

The graph of the squaring function f (x) = x2

is a U-shaped curve with the following characteristics.

• The domain of the function - all real numbers.

• The range - all nonnegative real numbers.

• The function is even : f(-x) = f(x) • The graph has an intercept at (0, 0).

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Linear and Squaring Functions

• The graph is decreasing on the interval ( , 0) and increasing on the interval (0, )

• The graph is symmetric with respect to the y-axis.

• The graph has a relative minimum at (0, 0).

Figure 1.68

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Cubic, Square Root, and Reciprocal Functions

1. Cubic function f (x) = x3

• The domain - all real numbers.

• The range - all real numbers.

• The function is odd : f(-x) = - f(x)

f(-x) = (-x)3 = - x3 = - f(x)

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The graph of the cubic function

• The graph is increasing on the interval ( , ).

• The graph is symmetric with respect to the origin.

Cubic function

Figure 1.69

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Square Root Function

2. The graph of the square root function f (x) =

• The domain – all nonnegative real numbers.

• The range - all nonnegative real numbers.

• The graph has an intercept at (0, 0).

• The Graph is increasing on the interval (0, )

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The Graph of Square Root Function

Square root function

Figure 1.70

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Reciprocal Function

3. The graph of the reciprocal function f (x) = has the following characteristics.

• The domain of the function is( , 0) (0, )

• The range of the function is ( , 0) (0, )

• The function is odd : f(-x) = - f(x)

• The graph does not have any y or x intercepts.

• The graph is decreasing on the intervals ( , 0) and (0, ).

• The graph is symmetric with respect to the origin.

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The graph of the Reciprocal Functions

.

Reciprocal function

Figure 1.71

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Step and Piecewise-Defined Functions

Step functions: Functions whose graphs resemble sets of stair steps

The most famous of the step functions is the greatest integer function, which is denoted by and defined as

f (x) = = the greatest integer less than or equal to x.

Some values of the greatest integer function are as follows.

= (greatest integer –1) = –1

= (greatest integer ) = –1

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Step and Piecewise-Defined Functions

= (greatest integer ) = 0

= (greatest integer 1.5) = 1

The graph of the greatest integer function

f (x) =

• The domain of the STEP function - all real numbers.

• The range of the STEP function - all integers.

• The graph has a y-intercept at (0, 0) and x-intercepts in the interval [0, 1).

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Step Function GRAPH

Figure 1.72

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Example 2 – Evaluating a Step Function

Evaluate the function when x = –1, 2 and

f (x) = + 1

Solution:

For x = –1, the greatest integer –1 is –1, so

f (–1) = + 1

= –1 + 1

= 0

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Example 2 – Solution

For x = 2, the greatest integer 2 is 2, so

f (2) = + 1

= 2 + 1

= 3.

For x = , the greatest integer is 1, so

= 1 + 1

= 2

cont’d

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Example 2 – Solution

You can verify your answers by examining the graph of

f (x) = + 1 shown in Figure 1.73.

cont’d

Figure 1.73

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Parent Functions

The most commonly used functions in algebra.

(a) Constant Function (b) Identity Function (c) Absolute Value Function

(d) Square Root Function

Figure 1.75

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Parent Functions

(e) Quadratic Function

(f) Cubic Function

(g) Reciprocal Function

(h) Greatest Integer Function

Figure 1.75