Copyright © 2011 Pearson, Inc.

1.3Twelve Basic

Functions

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What you’ll learn about

What Graphs can Tell Us Twelve Basic Functions Analyzing Functions Graphically

… and whyAs you continue to study mathematics, you will find that the twelve basic functions presented here will come up again and again. By knowing their basic properties, you will recognize them when you see them.

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The Identity Function

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The Squaring Function

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The Cubing Function

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The Reciprocal Function

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The Square Root Function

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The Exponential Function

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The Natural Logarithm Function

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The Sine Function

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The Cosine Function

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The Absolute Value Function

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The Greatest Integer Function

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The Logistic Function

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Example Looking for Domains

One of the functions has domain the set of all reals except 0.

Which function is it?

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Solution

One of the functions has domain the set of all reals except 0.Which function is it?

The function y =1/ x has a vertical asymptote at x=0.

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Example Analyzing a Function Graphically

Graph the function y =−x2 −1.Then answer the following questions.(a) On what interval is the function increasing? decreasing?(b) Is the function even, odd, or neither?(c) Does the function have any extrema?(d) How does the graph relate to the graph of the basic

function y=x2 ?

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Solution

(a) Increasing on −∞,0( ] and decreasing 0,∞[ )(b) Symmetric with respect to the y-axis, but not the origin so the function is even.(c) Maximum value of −1 at x=0

(d) The graph is the graph of y=x2 reflected over the x-axis and translated 1 unit down.

Graph the function y =−x2 −1.

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Quick Review

Evaluate the expressions without using a calculator.

1. −43.21

2. π −4

3. ln 1

4. e0

5. 44( )

4

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Quick Review Solutions

Evaluate the expressions without using a calculator.

1. −43.21 =43.21

2. π −4 =4−π

3. ln 1=0

4. e0 =1

5. 44( )

4

=4