Lial/Hungerford/Holcomb/Mullin: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All right reserved.
Copyright ©2015 Pearson Education, Inc. All right reserved. Chapter 3 Functions and Graphs Copyright ©2015 Pearson Education, Inc. All right reserved.
Copyright ©2015 Pearson Education, Inc. All right reserved. Section 3.1 Functions Copyright ©2015 Pearson Education, Inc. All right reserved.
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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Each of the given equations defines y as a function of x. Find the domain of each function. (a) Solution: Any number can be raised to the fourth power, so the domain is the set of all real numbers, which is sometimes written as (b) Solution: For y to be a real number, must be nonnegative. This happens only when —that is, when So the domain is the interval (c) Solution: Because the denominator cannot be 0, and the domain consists of all numbers in the intervals, Copyright ©2015 Pearson Education, Inc. All right reserved.
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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 3.2 Graphs and Functions Copyright ©2015 Pearson Education, Inc. All right reserved.
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Graph the absolute-value function, whose rule is Example: Graph the absolute-value function, whose rule is Solution: The absolute value function can be defined as the piecewise function So the right half of the graph (that is, where ) will consist of a portion of the line . It can be graphed by plotting two points, say The left half of the graph (where ) will consist of a portion of the line which can be graphed by plotting Copyright ©2015 Pearson Education, Inc. All right reserved.
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Applications of Linear Functions Section 3.3 Applications of Linear Functions Copyright ©2015 Pearson Education, Inc. All right reserved.
Copyright ©2015 Pearson Education, Inc. All right reserved. Example: An anticlot drug can be made for $10 per unit. The total cost to produce 100 units is $1500. (a) Assuming that the cost function is linear, find its rule. Solution: Since the cost function is linear, its rule is of the form We are given that m (the cost per item) is 10, so the rule is To find b, use the fact that it costs $1500 to produce 100 units which means that So the rule is (b) What are the fixed costs? Solution: The fixed costs are Copyright ©2015 Pearson Education, Inc. All right reserved.
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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 3.4 Quadratic Functions Copyright ©2015 Pearson Education, Inc. All right reserved.
Graph each of these quadratic functions: Example: Graph each of these quadratic functions: Solution: In each case, choose several numbers (negative, positive, and 0) for x, find the values of the function at these numbers, and plot the corresponding points. Then connect the points with a smooth curve to obtain the graphs. Copyright ©2015 Pearson Education, Inc. All right reserved.
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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 3.5 Polynomial Functions Copyright ©2015 Pearson Education, Inc. All right reserved.
Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Graph Solution: First, find several ordered pairs belonging to the graph. Be sure to choose some negative x-values, and some positive x-values in order to get representative ordered pairs. Find as many ordered pairs as you need in order to see the shape of the graph and draw a smooth curve through them to obtain the graph below. Copyright ©2015 Pearson Education, Inc. All right reserved.
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Copyright ©2015 Pearson Education, Inc. All right reserved. Section 3.6 Rational Functions Copyright ©2015 Pearson Education, Inc. All right reserved.
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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Graph Solution: Find the vertical asymptotes by setting the denominator equal to 0 and solving for x: Factor. Set each term equal to 0. Solve for x. Since neither of these numbers makes the numerator 0, the lines are vertical asymptotes of the graph. Copyright ©2015 Pearson Education, Inc. All right reserved.
Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Graph Solution: The horizontal asymptote can be determined by dividing both the numerator and denominator of by (the highest power of x that appears in either one). When is very large, the fraction is very close to 0, so the denominator is very close to 1 and is very close to 2. Hence, the line is the horizontal asymptote of the graph. Copyright ©2015 Pearson Education, Inc. All right reserved.
Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Graph Solution: Using this information and plotting several points in each of the three regions defined by the vertical asymptotes, we obtain the desired graph. Copyright ©2015 Pearson Education, Inc. All right reserved.
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