SECTION 1.2
Functions
71
(a) Does the point (1, 1) lie on the graph? Do the
coordinates satisfy the equation?
(b) What are the coordinates of point R? Do they
satisfy the equation?
(c) If the point P, with coordinates (a, b), is on the
graph, how are a and b related?
(d) What is the x-value of the point whose
y-coordinate is 0? Does this value of x satisfy the
equation 0 = 2x??
(d) Does f(x+h) = f(x) + h?
f(x +h)-f(x)
(e) Find
and simplify.
h
22. Let f(x) = 3x2 - 6x and h +0.
(a) Is f(3 + 2) = f(3) + 2?
(b) Find f(x + h).
(c) Does f(x + h) = f(x) +h?
(d) Does f(x + h) = f(x) + f(h)?
f(x + h)-f(x)
(e) Find
and simplify.
h
23. If f(x) = x - 2x' and h + 0, find the following and
simplify.
f(x+h)-f(x)
(a) f(x+h)
(b)
h
24. If f(x) = 2x - x + 3 and h = 0, find the following
and simplify.
f(x+h)-f(x)
(a) f(x + h)
(b)
h
25. If y = f(x) has the graph in Problem 5, find the
following
(a) f(9)
(b) f(5)
26. Suppose y = g(x) has the graph in Problem 8.
(a) Find g(0)
(b) How many x-values in the domain of this func-
tion satisfy g(x) = 0?
27. The graph of y = x - 4x is shown below.
State the domain and range of each of the functions in
Problems 29-32.
29. y = x + 4
30. y=x-1
31. y = V x + 4
32. y= Vr? + 1
In Problems 33-36, a function and its graph are given. In
each problem, find the domain.
Vx-1
x+1
33. f(x) =
34. f(x) =
X-2
Vx+ 3
10-
10+
THPT
4 6 8 10 12
2 4 6 8 10
y=x2-4r
S +
P(a,b)
3 4 5
35. f(x) = 4 + 49 – x2
36. f(x) = -2- V9 - x?
-2+
R
(a) What are the coordinates of the point Q? Do they
satisfy the equation?
(b) What are the coordinates of R? Do they satisfy the
equation?
(c) he coordinat of the point P the graph are
(a, b), how are a and b related?
(d) What are the x-values of the points on the graph
whose y-coordinates are 0? Are these x-values
solutions to the equation x - 4x = 0?
28. The graph of y = 2x is shown below.
HH
+ 2 4 6 8
P(a,b)
For f(x) and g(x) given in Problems 37-40, find
(a) (f+g)(x)
(b) (-3)(x)
(c) fg)(x)
(d) (f/g)(x)
37. f(x) = 3x g(x) = x
38. f(x) = x g(x) = 1/2
39. f(x) = 2x
g(x) = x2
40. f(x) = (x - 1) g(x) = 1 - 2x
y = 2x2
R
70
CHAPTER 1
Linear Equations and Functions
✓ CHECKPOINT
6. If f(x) = 1 – 2x and g(x) = 3x”, find the following.
(a) (g-f)(x) (b) (fog)(x) (c) (fºg)(x)
(e) (ff)(x) = f(f(x))
(d) (gºf)(x)
✓ CHECKPOINT
ANSWERS
1. Independent variable is x; dependent variable is y.
2. f(1) = 3
3. f(-2) = 9
4. f(x + h) = 2(x + 2xh + h)
5. All real numbers except x = -1; f(x) is undefined when x = -1.
6. (a) (g-f)(x) = 3x2 + 2x - 1
(b) (fog)(x) = 3x2 - 6x
(c) (fºg)(x) = 1 - 6x?
(d) (gºf)(x) = 3(1 - 2x)
(e) (ff)(x) = 4x - 1
| EXERCISES | 1.2
7.
In Problems 1 and 2, use the values in the following
table.
4+
х
- 7
-1
0
4.2
9
11
14
18
22
3
5
у
0
0
1
9
11
35 22 22 60
HAHAHHX
-8 - 4 V 6 8 10
4
-6 +
-10+
1. (a) Explain why the table defines y as a function of x.
(b) State the domain and range of this function.
(c) If the table expresses y = f(x), find f(0) and f(11).
2. (a) If the function defined by the table is denoted by f,
so that y = f(x), is f(9) an input or an output off?
(b) Does the table describe x as a function of y?
Explain.
In Problems 3 and 4, are the relations defined by the
tables functions? Explain why or why not and give the
domain and range.
1 2 3 8 9
-4 -4 5 16 5
3.
X
y
х
-1
1
3
1
9
у
0
2
4
6
Problems 9-12, determine if the equation represents y
as a function of x.
9. y = 3x3
10. y = 6x2
11. y = 3x
12. y2 = 10x
13. If R(x) = 8x – 10, find the following.
(a) R(O) (b) R(2) (c) R(-3) (d) R(1.6)
14. If f(x) = 17 - 6x, find the following.
(a) f(-3) (b) f(1) (c) f(10) (d) f)
15. If C(x) = 4x - 3, find the following.
(a) C(O) (b) C(-1) (c) C(-2) (d) C(-2)
16. If h(x) = 3x2 – 2x, find the following.
(a) h(3) (b) h(-3) (c) h(2) (a) h(1)
17. If h(x) = x - 24 – x), find the following.
(a) h(-1) (b) h(0) (c) (6) (d) h(2.5)
18. If R(x) = 100x - x, find the following.
(a) R(1) (b) R(10) (c) R(2) (d) R(-10)
19. If f(x) = x - 4/x, find the following.
(a) f(-) (b) f(2) (c) f(-2)
20. If C(x) = (x2 - 1)/x, find the following.
(a) C(1) (b) () (c) C(-2)
21. Let f(x) = 1 + x + x' and h +0.
(a) Is f(2 + 1) = f(2) + f(1)?
(b) Find f(x + h).
(c) Does f(x + h) = f(x) + f(h)?
In Problems 5-8, determine whether each graph repre-
sents y as a function of x. Explain your answer.
5.
6.
12+ (9.10)
(5.6)
-6
HHHHH
2 4 6 8 10 12
(2.2)
-6+
82
CHAPTER 1
Linear Equations and Functions
Solution
(a) To put the equation in slope-intercept form, we must solve it for y.
2y = -x + 8
or
y =
+4
Thus the slope is and the y-intercept is 4.
(b) First we plot the y-intercept point (0,4). Because the slope is { =, moving 2
units to the right and down 1 unit from 0, 4) gives the point (2, 3) on the line.
A third point (for a check) is plotted at (4,2). The graph is shown in Figure 1.16.
y = -x + 4
(0.4)
(2,3)
2
(4,2)
2+
1
Figure 1.16
It is also possible to graph a straight line if we know its slope and any point on the line;
we simply plot the point that is given and then use the slope to plot other points.
The following summarizes the forms of equations of lines.
Forms of Linear Equations
General form: ax + by+c=0
Point-slope form: y - y = m(x - x)
Slope-intercept form: y = mx + b
Vertical line: x= a
Horizontal line: y = b
CHECKPOINT
4. Write the equation of the line that has slope - and passes through (4, -6).
5. What are the slope and y-intercept of the graph of x = - 4y + 1?
✓ CHECKPOINT
ANSWERS
1
1. m=
2
2. Ifm = 0, then the line is horizontal. If m is undefined, then the line is vertical.
1
3. (a) m =
(b) m = 5
5
4. y = -x-3.
5. m= -1; y-intercept is 4.
| EXERCISES 1.3
In Problems 1-4, find the intercepts and graph the
functions.
1. 3x + 4y = 12
2. 6x - 5y = 90
3. 5x – 8y = 60
4. 2x - y + 17 = 0
In Problems 5-10, find the slope of the line passing
through the given pair of points.
5. (22, 11) and (15,-17)
6. (-6, -12) and (-18, -24)
SECTION 1.3
Linear Functions
83
39.
7. (3,-1) and (-1,1)
8. (-5,6) and (1, -3)
9. (3, 2) and (-1,2)
10. (-4, 2) and (-4,-2)
11. If a line is horizontal, then its slope is
12. If a line is vertical, then its slope is
13. What is the rate of change of the function whose graph
is a line passing through (3, 2) and (-1, 2)?
14. What is the rate of change of the function whose graph
is a line passing through (11, -5) and (-9, - 4)?
-16
40.
In Problems 15 and 16, for each given graph, determine
whether each line has a slope that is positive, negative, 0,
or undefined.
15
16.
(a) 10+
(b)
10+
(b)
5+
8
2
(a)
-107
In Problems 17-24, find the slope and y-intercept and
then graph each equation.
7 1
4 1
17. y =
18. y = x +
3 2
19. y = 3
20. y = -2
21. X = -8
22. X = -1/2
23. 2x + 3y = 6
24. 3x – 2y = 18
4
In Problems 41-44, determine whether the following
pairs of equations represent parallel lines, perpendicular
lines, or neither of these.
41. 3x + 2y = 6; 2x - 3y = 6
42. 5x – 2y = 8; 10x – 4y = 8
43. 6x - 4y = 12; 3x - 2y = 6
4
44. 5x + 4y = 7; y = x + 7
45. Write the equation of the line passing through
(-2,- 7) that is parallel to 3x + 5y = 11.
46. Write the equation of the line passing through (6,-4)
that is parallel to 4x - 5y = 6.
47. Write the equation of the line passing through (3, 1)
that is perpendicular to 5x – 6y = 4.
48. Write the equation of the line passing through
(-2,-8) that is perpendicular to x = 4y + 3.
In Problems 25-28, write the slope-intercept form of
the equation of the line that has the given slope and
y-intercept. Then graph the line.
25. Slope and y-intercept - 3
26. Slope 4 and y-intercept 2
27. Slope -2 and y-intercept
28. Slope - and y-intercept - 1
APPLICATIONS
In Problems 29-34, write the equation of the line that
passes through the given point and has the given slope.
Then the line.
29. (2,0) with slope - 5 30. (1, 1) with slope -
31. (-1, 4) with slope - 32. (3,-1) with slope 1
33. (-1, 1) with undefined slope
34. (1, 1) with 0 slope
49. Depreciation A $360,000 building is depreciated by
its owner. The value y of the building after x months o
use is y = 360,000 - 1500x.
(a) Graph this function for x 20.
(b) How long is it until the building is completely
depreciated (its value is zero)?
(c) The point (60, 270,000) lies on the graph. Explain
what this means.
50. U.S population Using Social Security Administration
data for selected years from 1950 and projected
to 2050, the U.S. population (in millions) can be
described by
p(t) = 2.537 + 162.2
In Problems 35-40, write the equation of each line
described or shown.
35. Through (3, 2) and (-1,-6)
36. Through (-4, 2) and (2, 4)
37. Through (7,3) and (-6,2)
38. Through (10, 2) and (8,7)
72
CHAPTER 1 Linear Equations and Functions
For f(x) and g(x) given in Problems 41-44, find
(a) (fog)(x)
(b) (gºf)(x)
(c) f(f(x))
(d) f'(x)=(f.f)(x)
41. f(x) = (x - 1) g(x) = 1 - 2x
42. f(x) = 3x g(x) = x - 1
43. f(x) = 2x g(x) = x4 + 5
44. f(x) ) = g(x) = 4x + 1
(b) Find f(10) and write a sentence that explains its
meaning.
(c) Is f(5+5) = f(5) + f(5)? Explain.
47. Social Security benefits The figure below gives the
monthly Social Security benefits for persons whose
benefit would be $1000 at a full retirement age of 66 as
a function of the age these beneficiaries start receiving
benefits. The figure shows the benefits, y = f(x), as a
function of x, for 62 $x = 70.
(a) Find f(64) and f(67).
(b) Find f(68) and give its meaning.
(c) Find f(66) - f(62) and gives its meaning.
APPLICATIONS
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