Copyright © 2019 Pearson Education, Inc. Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 12e Finite Mathematics with Applications 12e Copyright © 2019 Pearson Education, Inc. Slide 1 ALWAYS LEARNING

Chapter 3 Functions and Graphs

Section 3.1 Functions

Figure 3.1

Example: Solution: Solution: Solution: 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: For y to be a real number here, we must have which is equivalent to So the domain interval is

Example: Solution: Solution: Each of the given equations defines y as a function of x. Find the domain of each function. (d) Solution: We can take the cube root of any real number, so the domain is the set of all real numbers (e) Solution: Because the denominator cannot be 0, and the domain consists of all numbers in the intervals,

Example: Each of the given equations defines y as a function of x. Find the domain of each function. (f) Solution: The numerator is defined only when The domain cannot contain any numbers that make the denominator0- that is, the numbers that are solutions of Therefore, the domain consists of all nonnegative real numbers except 1 and 2.

Section 3.2 Graphs of Functions

Figure 3.5

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 Figure 3.6 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

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 Figure 3.6 Many real world problems can be modeled using piecewise-defined functions. For example, every employee pays Social Security and Medicare taxes, and these formulas are piecewise-defined. The costs for many products and services are piecewise-defined due to either volume discounts or higher rates for excessive usage.

Figure 3.7

Figure 3.8

Figure 3.9

Applications of Linear Functions Section 3.3 Applications of Linear Functions

Example: Solution: Solution: 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 C(x)=10x+b So the rule is (b) What are the fixed costs? Solution: The fixed costs are

Figure 3.19

Figure 3.22

Quadratic Functions and Applications Section 3.4 Quadratic Functions and Applications

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 in Figure 3.23.

Figure 3.29

Figure 3.30

Figure 3.31

Figure 3.32

Figure 3.33

Section 3.5 Polynomial Functions

Example: Solution: Graph Make a table of data with ordered pairs belonging to the graph, as in Figure 3.34a. 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. Then plot the ordered pairs and draw a smooth curve through them to obtain the graph in Figure 3.34b.

Section 3.6 Rational Functions

Figure 3.48

Example: Solution: Graph Find the vertical asymptotes by setting the denominator equal to 0 and solving for x: Set denominator equal to 0. 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.

Example: Solution: Graph 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 and is very close to 2. Hence, the line is the horizontal asymptote of the graph.

Example: Solution: Graph Using this information and plotting several points in each of the three regions defined by the vertical asymptotes, we obtain Figure 3.51.