College Algebra

Functions and Graphs(Chapter1)

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Objectives

Chapter1

Cover the topics in Section ( 1-4):Graphs and properties

1. Basic Ideas

2. Determine the intervals on which a function is increasing, decreasing or constant by looking at a graph.

3. Determine if a function is even, odd, or neither by looking at a graph.

4. Determine if a function is even, odd, or neither given an equation.

5. Apply the greatest integer function to any given number. 

After completing this section, you should be able to:

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Graphs and properties

Chapter1

In other words, a function is increasing in an interval if it is going up left to right in the entire interval.

Increasing A function is increasing on an interval if for any X1 and X2 in the interval,

where X1<X2 , then f(X1) < f(X2) .

Below is an example where the function is increasing over the interval           .  Note how it is going up left to right in the interval           .

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Graphs and properties

Chapter1

In other words, a function is decreasing in an interval if it is going down left to right in the entire interval.

Below is an example where the function is decreasing over the intervalNote how it is going down left to right in the interval

Decreasing A function is decreasing on an interval if for any X1 and X2 in the

interval, where X1<X2 , then f(X1) > f(X2) .

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Graphs and properties

Chapter1

In other words, a function is constant in an interval if it is horizontal in the entire interval.

Below is an example where the function is constant over the interval Note how it is a horizontal line in the interval

Constant A function is constant on an interval if for any X1 and X2 in the interval,

where X1<X2 , then f(X1)= f(X2) .

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Graphs and properties

Chapter1

a) Increasing

b) Decreasing (2, 3)

c) Constant (-5, 2)

Example 1: Use the graph to determine intervals on which the function is a) increasing, if any, b) decreasing, if any, and c) constant, if any.

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Graphs and properties

Chapter1

a) Increasing

b) Decreasing

c) Constant ------

Example 2: Use the graph to determine intervals on which the function is a) increasing, if any, b) decreasing, if any, and c) constant, if any.

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Graphs and properties

Chapter1

In other words, a function is even if replacing x with -x does NOT change the original function.

In terms of looking at a graph, an even function is symmetric with respect to the y-axis.  In other words, the graph creates a mirrored image across the y-axis. The graph below is a graph of an even function.  Note how it is symmetric about the y-

axis .

Even Function A function is even if for all x in the domain of f

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Graphs and properties

Chapter1

In other words, a function is odd if replacing x with -x results in changing every sign of every term of the original function.

In terms of looking at a graph, an odd function is symmetric with respect to the origin.  In other words, the graph creates a mirrored image across the origin .

 The graph below is a graph of an odd function.  Note how it is symmetric about the origin .

Odd Function A function is odd if for all x in the domain of f

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Graphs and properties

Chapter1

To determine if this function is even, odd, or neither, we need to replace x with -x and compare f(x) with f(-x):

Example 3: Determine if the function is even, odd or neither?

The function is not even

The function is not odd

Final answer: The function is neither even nor odd.

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Graphs and properties

Chapter1

To determine if this function is even, odd, or neither, we need to replace x with -x and compare g(x) with g(-x):

Example 4: Determine if the function is even, odd or neither?

Final answer: The function is even.

             .

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Graphs and properties

Chapter1

To determine if this function is even, odd, or neither, we need to replace x with -x and compare f(x) with f(-x):

Example 5: Determine if the function is even, odd or neither?

Final answer: The function is odd.

The function is not even

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Graphs and properties

Chapter1

For example, int(5) = 5, int(5.3) = 5, int(5.9) = 5, because 5 is the greatest integer that is less than or equal to 5, 5.3, and 5.9 .The basic graph of the function f(x) = int(x) is :

Greatest Integer Function int(x)

Greatest integer that is less than or equal to x.

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Graphs and properties

Chapter1

We need to ask ourselves, what is the greatest integer that is less than or equal to 7.92? If you said 7, you are correct.

Final answer: 7

Example 6: If f(x) = int(x), find the functional value f(7.92).

We need to ask ourselves, what is the greatest integer that is less than or equal to 7.92?

Final answer: 7

Example 6: If f(x) = int(x), find the functional value f(7.92).

We need to ask ourselves, what is the greatest integer that is less than or equal to -3.25? Be careful on this one.  We are working with a negative number.  -3 is not a correct answer because -3 is not less than or equal to -3.25, it is greater than -3.25.

Final answer: -4

Example 7:If f(x) = int(x), find the functional value f(-3.25).