1.1 Functions 1
Exercises 1.1
1. At any given instant in time, you can have only one weight. For example, at age 19 years, 3 months, 2 days, 11 hours, 2 minutes, 27 seconds you have only one weight. Thus your weight is a function of your age.
3. Temperature is a function of the time of day since at any time of the day when the temperature is measured, the measured temperature will be a single value.
5. The input value 182 has two different output values (32 and 47). Therefore, the number of salmon in a catch is not a function of the number of fish caught.
7.
The total cost of four pairs of shoes is $159.80.
9.
Two seconds after he jumped, the cliff diver is 56 feet above the water.
11. As shown in the table, . In the fourth quarter since December 1999, shares in the tortilla company earned $0.06 per share.
13. As seen in the graph, .
On November 8, 2001, the closing stock price of the computer company was approximately $18.72. (One drawback of reading a graph is that it is difficult to be precise.)
15. It appears that near , the graph goes vertical. A vertical line drawn at that point would touch the graph in multiple locations. However, if the graph doesn’t actually go vertical near , then it is a function. One drawback of reading a graph is that it is sometimes difficult to tell if the graph goes vertical or not.
17. Since any vertical line drawn will touch the graph exactly once, the graph is a function.
19. Since any vertical line drawn will touch the graph exactly once, the graph is a function.
21.
23.
25. ;
From the graph, it appears that at . We calculate the exact value of y algebraically:
27. ;
From the graph, it appears that at . We calculate the exact value of y algebraically:
But division by zero is not a legal operation. Therefore, the function is not defined when . Graphically speaking, there is a “hole” in the graph at .
29. The function has the domain of all real numbers.
31. The function has a domain of all real numbers, since no value of r will make the denominator equal to zero.
33. The function is undefined when the denominator equals zero.
The domain of the function is all real numbers except . That is, .
35. The function is undefined when the radicand is negative. So
The domain of the function is all real numbers greater than or equal to –1. That is, .
37. The function is undefined when the radicand is negative or the denominator is equal to zero.
The denominator is always positive. The domain of the function is the set of all real numbers greater than or equal to –3. That is, .
39. Since it doesn’t make sense to sell a negative number of bags of candy, n is nonnegative. The domain of the candy profit function is the set of whole numbers. That is, .
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1.2 Linear Functions 7
41. Using our current calendaring system, the domain of the function is the set of whole numbers between 1 and the current year. For example, in 2006, the domain of the function is . (It is impossible to calculate the average height of z of a person yet to be born in future years.)
43. The function is undefined whenever the denominator is equal to zero.
The domain of the function is all real numbers except and .
45. Yes. Even though the domain value of is listed twice in the table, it is linked with the same range value, .
Exercises 1.2
1. The slope of the line passing through (2, 5) and (4, 3) is
3. The slope of the line passing through (1.2, 3.4) and (2.7, 3.1) is
5. The slope of the line passing through (2, 2) and (5, 2) is
7. y-intercept:
x-intercept:
9. y-intercept:
x-intercept:
11.
y-intercept:
x-intercept:
13. The slope of the line passing through (2, 5) and (4, 3) is
The slope-intercept form of the line is . The standard form of the line is. A point-slope form of the line is .
15. The slope of the line passing through (1.2, 3.4) and (2.7, 3.1) is
The slope-intercept form of the line is . The standard form of the line is typically written with integer coefficients. Therefore,
is the standard form of the line.
A point-slope form of the line is .
17. The slope of the line passing through and (5, 2) is
Since the slope is equal to zero, this line is a horizontal line. The slope-intercept form of the line is and is commonly written as . The standard form of the line is and is also often written as . A point-slope form of the line is .
19.
21.
23.
25.
27. and are two points on the line. The slope of the line is
The slope-intercept form of the line is .
29. This is a vertical line with x-intercept . The equation of the line is .
31. A table containing exactly two points each with different x-values will always represent a linear function.
year 2000 dollars)
1989 / 18,593
1999 / 28,525
Source: www.census.gov
The slope is given by
Between 1989 and 1999, the U.S. average personal income (in year 2000 dollars) increased by an average of $993.20 per year.
33.
Months (since Sep 01) / Take Home Pay (dollars)0 / 3167.30
1 / 4350.31
Source: Employee pay stubs
The slope is given by
Between September 2001 and October 2001, the employee’s take home pay increased at a rate of $1183.01 per month.
35. If the table of data represents a linear function then a linear function passing through two of the points will also pass through all other points in the table.
500 / $18.75
700 / $26.25
900 / $33.75
1000 / $37.50
Source: www.dnr.metrokc.gov
The slope of the line passing through and is
The equation of the line passing through these points is given by
We evaluate the linear equation at and .
These results match the table data. The data table does represent a linear function. It costs an average of $0.0375 per pound to dispose of clean wood.
37. Let x be the number of servings of WheatiesTM and y be the grams of fiber consumed. We have
since each serving of cereal contains 2.1 grams of fiber and the banana contains 3.3 grams of fiber. We must solve
In order to consume 8-grams of fiber, you would need to eat 3 servings (cups) of Wheaties along with the large banana.
39. The slope of the line is
Therefore, the line is a vertical line. Although the y-values change, every point on a vertical line has the same . The x-value of each of the points is. The equation of the vertical line is .
41. The slope-intercept form is and the point-slope form is . For a vertical line, the slope m is undefined and thus may not be substituted into either of the two forms.
43. Vertical lines are not functions since they fail the Vertical Line Test. However, all non-vertical lines are functions.
45. The slope of the line is 0 since may be written as . The slope of a line perpendicular to this line has slope
Therefore, the perpendicular line is a vertical line. Since all points on a vertical line have the same x-values, the equation of the line is .
47. We can find the x-intercept by plugging in and the y-intercept by plugging in .
The x-intercept is .
The y-intercept is .
The slope of the line is
49. We will first find the equation of the line passing through and .
So . We’ll now plug in the point and solve for b.
In order for the table of data to represent a linear function, b must equal 23.
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1.3 Linear Models 13
51. The point of intersection of the lines is . This point is the only point that satisfies both of the linear equations.
Exercises 1.3
1. a. The scatter plot shows that the data is near linear.
b. Using linear regression on the TI83 Plus, we determine
The equation of the line of best fit is.
c. means that the Harbor Capital Appreciation Fund share price is dropping at a rate of $6.29 per month.
The y-intercept means that in month 0 of 2000 the fund price was $110.94. Since the months of 2000 begin with 1 not 0, the y-intercept does not represent the price at the end of January. It could, however, be interpreted as being the price at the end of December1999.
d. This model is a useful tool to show the trend in the stock price between October and December2000. Since stock prices tend to be volatile, we are somewhat skeptical of the accuracy of data values outside of that domain.
3. a. The scatter plot shows that the data is near linear.
b. Using linear regression on the TI83 Plus, we determine
The equation of the line of best fit is.
c. means that Washington State public university enrollment is increasing by about 1166 students per year. means that, according to the model, Washington State public university enrollment was 80,887 in 1990.
d. The model fits the data extremely well as shown by the graph of the line of best fit.
This model could be used by Washington State legislators and university administrators in budgeting and strategic planning.
5. a. The scatter plot shows that although the data are not linear, they are nonincreasing.
b. Using linear regression on the TI83 Plus, we determine
The equation of the line of best fit is.
c. means that the per capita income ranking of North Carolina is changing at a rate of 3.4 places per year. That is, the state is moving up in the rankings by about 3 places per year.
The y-intercept means that in 1995 (year 0), North Carolina was ranked 40th out of the 50 states. (Only positive whole number rankings make sense.)
d. This model is not a highly accurate representation of the data as shown by the graph of the line of best fit so it should be used with caution.
However, an incumbent government official could use the model in a 2000 reelection campaign as evidence that the state’s economy had improved during his or her tenure in office. The official might also use the model to claim that the trend of improvement will continue if he or she is reelected.
7. a. The minimum fees and the rounding of the weight of the trash make this particular function somewhat complicated. We will first find how much trash may be disposed of for the minimum fees.
The amount of trash that can be disposed of for the minimum disposal fee is
The largest multiple of twenty that is less than or equal to 332.6 is 320. Therefore, a rounded weight of 320 pounds of trash may be disposed of for the minimum disposal fee.
The amount of trash that can be disposed of for the minimum moderate risk waste fee is
The largest multiple of twenty that is less than or equal to 766.3 is 760.
For each of the following functions, we let x represent the weight of the trash rounded to the nearest 20 pounds.
The cost of disposing 320 pounds of trash or less (including tax) is
The cost of disposing between 340 and 760 pounds of trash (including tax) is
The cost of disposing of 780 pounds of trash or more is
Combining the individual functions into a single piecewise cost function we have
For values of , is directly proportional to x.
b. We must first round the weight of the trash to the nearest 20 pounds. One way to do this is to divide the weight of the trash by 20, round the number to the nearest whole number, and multiply the result by 20. That is,
Since this weight is below 320 pounds, the minimum $15.25 fee will be charged.
c. How much will it cost to drop off 513 pounds of trash?
d. The cost per pound is lowest when 780 or more pounds of trash are disposed of. As a construction company, we would try to keep the weight of our trash deliveries at or above 780 pounds.
9. a. Let C be the total cost (in dollars) of using t anytime minutes of phone time during a given month. (If a fraction of a minute is used, t is the next whole number value. For example, if 3.2 anytime minutes are used then .) The $29.99 fee covers the first 200 minutes. The $0.40 per minute fee is only charged on the minutes used after the first 200 minutes. Thus we have
which may be rewritten as
b. For the Qwest plan we have
For the Sprint plan we have
The Sprint plan is the best deal for a customer who uses 300 anytime minutes. If a plan was selected that offered 300 anytime minutes for a higher basic fee, it would likely cost the customer even less.
11. a. Let t be the production year of a Toyota Land Cruiser 4-Wheel DriveTM and V be the value of the vehicle in 2001. We have and . We must find the linear function passing through these points. We have
so . We solve for b using the point .
The linear model is .
b. We have
According to the model, a 1992 Toyota Land Cruiser was valued at $7610 in 2001.
c. The model substantially underestimated the value of a 1992 Land Cruiser. If we draw a scatter plot of the value of a 1992, 1995, and 2000 Land Cruisers in 2001, we see that the vehicle value function is not linear.
13. a. Let F be the number of grams of fat in x chef salads. We have
.
The number of fat grams is directly proportional to the number of chef salads.
b. We can consume up to 80 grams of fat.