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Lesson 2.1 Exercises, pages 90–96
A
4. a) Complete the table of values.
ⴚ4
ⴚ2
ⴚ1
0
1
2
4
y ⴝ ⴚx
4
2
1
0
ⴚ1
ⴚ2
ⴚ4
√
y ⴝ ⴚx
2
1.4
1
0
ⴚ
ⴚ
ⴚ
x
b) For each function in part a, sketch its graph then state its domain
and range.
For y ⴝ ⴚx: the domain is x ç ⺢; and the range is y ç ⺢.
√
For y ⴝ ⴚx: the domain is x ◊ 0; and the range is y » 0.
y x
√
y x
4
c) How are the graphs in part b related?
√
2
4
y
2
x
0
2
4
2
Both graphs pass through (ⴚ1, 1) and (0, 0). The graph of y ⴝ ⴚx is
above the graph of y ⴝ ⴚx between these points. Every value of y
√
for y ⴝ ⴚx is the square root (if it exists) of the corresponding
y-value on the graph of y ⴝ ⴚx.
5. For each graph of y = g(x) below:
i) Mark points where y = 0 or y = 1.
√
ii) Sketch the graph of y = g(x).
iii) Identify the domain and range of y =
a)
b)
y
4
2
4
0
2
y g(x)
√
y g(x)
x
4
4
√
g(x).
y
2
2 0
y g(x)
2
√
y g(x)
x
4
ii) Choose the point (3, 4) on
y ⴝ g(x). The
corresponding point on
√
y ⴝ g(x) is (3, 2). Draw a
curve through this point
and the marked points.
ii) Choose the point (5, 4) on
y ⴝ g(x). The corresponding
√
point on y ⴝ g(x) is (5, 2).
Draw a curve through this
point and the marked
points.
iii) Domain is: x » ⴚ1
Range is: y » 0
iii) Domain is: x » 1
Range is: y » 0
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c)
4
4
d)
y
y
y g(x)
2
√
y g(x)
2
2
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4
x
2
y g(x)
√
y g(x)
4
2
2
x
0
ii) Choose the point (ⴚ5, 4)
on y ⴝ g(x). The
corresponding point on
√
y ⴝ g(x) is (ⴚ5, 2). Draw
a curve through this point
and the marked points.
ii) Choose the point (ⴚ3, 4) on
y ⴝ g(x). The corresponding
√
point on y ⴝ g(x) is
(ⴚ3, 2). Draw a curve
through this point and the
marked points.
iii) Domain is: x ◊ ⴚ1
Range is: y » 0
iii) Domain is: x ◊ 1
Range is: y » 0
B
6. a) Use technology to graph each function. Sketch each graph.
i) y =
iii) y =
v) y =
√
2x + 3
√
√
ii) y =
√
-2x + 3
-2x - 3
iv) y =
√
0.5x + 4
-0.5x - 4
vi) y =
√
0.5x - 4
b) Given a linear function of the form f(x) ⫽ ax ⫹ b, a, b Z 0
√
i) For which values of a does the graph of y = f(x) open to
the right? Use examples to support your answer.
The graph opens to the right when a>0, as in part a, i, iv, and vi.
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√
ii) For which values of a does the graph of y = f(x) open to
the left? Use examples to support your answer.
The graph opens to the left when a<0, as in part a, ii, iii, and v.
7. a) Complete this table of values for y ⫽ x2 and y =
√
x2, then graph
the functions on the same grid.
ⴚ3
ⴚ2
ⴚ1
0
1
2
3
8
y ⴝ x2
9
4
1
0
1
4
9
6
yⴝ
3
2
1
0
1
2
3
x
√
x2
y
y x2
4
√
y x2
2
x
√
4
b) What other function describes the graph of y = x2?
Explain why.
√
The graph of y ⴝ x 2 is the√same as√the graph of y ⴝ 円 x円 . For any
2
2
2
2
number x, x ⴝ (ⴚx) ; both
root, which is 円x円 .
x and
2
0
2
4
(ⴚx) are equal to a positive
8. a) For the graph of each quadratic function y ⫽ f(x) below:
• Sketch the graph of y =
√
f(x).
• State the domain and range of y =
i)
4
ii)
y
16
x
6
4
2
4
Mark points where y ⴝ 0
or y ⴝ 1. Choose, then mark
another point on the graph
√
of y ⴝ f(x).
x
y ⴝ f(x)
ⴚ5
4
yⴝ
√
f(x)
2
Join all points with a smooth
curve.
Domain is: ⴚ7 ◊ x ◊ ⴚ3
Range is: 0 ◊ y ◊ 2
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y f(x)
8
0
√
y f (x)
x
4
2
y f(x)
y
12
2
√
y f (x)
8
√
f(x).
2
0
2
There are no points where
y ⴝ 0 or y ⴝ 1. Choose, then
mark other points on the graph
√
of y ⴝ f(x).
yⴝ
√
f(x)
x
y ⴝ f(x)
ⴚ2
12
⬟ 3.5
0
4
2
2
12
⬟ 3.5
Join all points with a smooth
curve.
Domain is: x ç ⺢
Range is: y » 2
2.1 Properties of Radical Functions—Solutions
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√
b) Choose one pair of functions y = f(x) and y = f(x) from
part a. If the domains are different and the ranges are different,
explain why.
Sample response: For part a, i, the domains are different because the
radical function only exists for those values of x where y » 0; while the
quadratic function exists for all real values of x. The ranges are different
because the value of y for the radical function can only be 0 or a
positive number; while the range of the quadratic function is all real
numbers less than or equal to 4, which includes all negative real
numbers.
9. Solve each radical equation by graphing. Give the solution to the
nearest tenth where necessary.
√
√
a) x - 5 = 2 x + 3
b) x = 4 - x + 2
Write the equation as:
√
xⴚ5ⴚ2 xⴙ3ⴝ0
Graph the related function:
√
f(x) ⴝ x ⴚ 5 ⴚ 2 x ⴙ 3
The zero is: 13
So, the root is: x ⴝ 13
√
c) 3 2x + 1 = x + 4
Write the equation as:
√
3 2x ⴙ 1 ⴚ x ⴚ 4 ⴝ 0
Graph the related function:
√
f(x) ⴝ 3 2x ⴙ 1 ⴚ x ⴚ 4
The approximate zeros are:
0.75735931 and 9.2426407
So, the roots are: x 0.8 and
x 9.2
4
Write the equation as:
√
xⴚ 4ⴚxⴚ2ⴝ0
Graph the related function:
√
f(x) ⴝ x ⴚ 4 ⴚ x ⴚ 2
The zero is: 3
So, the root is: x ⴝ 3
d) 1 +
√
√
x - 3 = 2x - 6
Write the equation as:
√
√
1 ⴙ x ⴚ 3 ⴚ 2x ⴚ 6 ⴝ 0
Graph the related function:
√
√
f(x) ⴝ 1 ⴙ x ⴚ 3 ⴚ 2x ⴚ 6
The approximate zero is:
8.8284271
So, the root is: x 8.8
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10. For the graph of each cubic function y = g(x) below:
√
• Sketch the graph of y = g(x).
√
• State the domain and range of y = g(x).
a)
b)
y
y g(x)
y
12
12
8
8
4
4
√
y g(x)
y g(x)
√
y g(x)
x
x
2
0
0
2
Mark points where
y ⴝ 0 or y ⴝ 1.
Identify and mark the
coordinates of other points
√
on the graph of y ⴝ g(x).
x
y ⴝ g(x)
ⴚ2
15
2
3
yⴝ
√
g(x)
2
Mark points where y ⴝ 0
or y ⴝ 1. Identify and mark
the coordinates of other
points on the graph of
√
y ⴝ g(x).
y ⴝ g(x)
3.9
ⴚ1
3
1.7
1.7
3
15
3.9
Join the points with 2 smooth
curves.
Domain is: x ◊ ⴚ1 or
1 ◊ x ◊ 3
Range is: y » 0
yⴝ
√
x
g(x)
Join the points with 2 smooth
curves.
Domain is: ⴚ2 ◊ x ◊ 0 or
x » 2
Range is: y » 0
11. a) Sketch the graph of a linear function y = g(x) for which
√
y = √g(x) is not defined. Explain how you know that
y = g(x) is not defined.
√
Sample response: For y ⴝ g(x) to be undefined, the value of
g(x) must always be negative; for example, a function is y ⴝ ⴚ3.
4
y
2
x
4
2
O
2
2
4
y 3
4
b) Sketch
√ the graph of a quadratic function y = f(x) for which
y = √f(x) is not defined. Explain how you know that
y = f(x) is not defined.
√
Sample response: For y ⴝ f(x) to be undefined, the value of f(x) must
always be negative; so the graph of y ⴝ f(x) must always lie beneath
the x-axis; for example, a function is y ⴝ ⴚ(x ⴚ 2)2 ⴚ 2.
y (x 2)2 2
y
x
4
2
0
2
4
2
4
6
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c) For every cubic function y = g(x), the function y =
Explain why.
√
g(x) exists.
Since the graph of every cubic function either begins in Quadrant 2 and
ends in Quadrant 4, or begins in Quadrant 3 and ends in Quadrant 1,
there is always part of the graph above the x-axis; that is, the function
has positive values, so its square root exists.
12. For each graph of y = g(x), sketch the graph of y =
a)
b)
y
4
√
g(x).
y
6
4
2
√2
y g(x)
y g(x)
√
y g(x)
4
2
2
0
2
yⴝ
√
g(x)
x
y ⴝ g(x)
ⴚ5
4
2
ⴚ3
2
1.4
1
2
1.4
3
4
2
4
6
y g(x)
4
Mark points where y ⴝ 0 or
y ⴝ 1. Identify and mark
the coordinates of other
points on the graph of
√
y ⴝ g(x).
x
2
2
x
4
0
Mark points where y ⴝ 0
and y ⴝ 1. Identify and mark
the coordinates of other
points on the graph of
√
y ⴝ g(x).
yⴝ
√
g(x)
x
y ⴝ g(x)
ⴚ4
4
2
4
2
1.4
Join the points with 2 smooth
curves.
Join the points with 2 smooth
curves, and a line segment.
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13. When a satellite is h kilometres above Earth, the time for one
complete orbit, t√
minutes, can be calculated using this formula:
-4
t = 1.66 * 10 (h + 6370)3
A communications satellite is to be positioned so that it is always
above the same point on Earth’s surface. It takes 24 h for this satellite
to complete one orbit. What should the height of the satellite be?
Substitute t ⴝ (24)(60), or 1440.
√
1440 ⴝ 1.66 ⴛ 10ⴚ4 (h ⴙ 6370)3
Write the equation with all the terms on one side.
√
1440 ⴚ 1.66 ⴛ 10ⴚ4 (h ⴙ 6370)3 ⴝ 0
Write a related function.
√
f(h) ⴝ 1440 ⴚ 1.66 ⴛ 10ⴚ4 (h ⴙ 6370)3
Graph the function, then determine the approximate zero, which is
35 848.513.
So, to the nearest kilometre, the satellite should be 35 849 km high.
C
√
3
f(x) without
using graphing√technology. What are the invariant points on the
graph of y = 3 f(x)?
14. Given the graph of y = f(x), sketch the graph of y =
√
√
√
y
Since 3 ⴚ1 ⴝ ⴚ1, 3 0 ⴝ 0, and 3 1 ⴝ 1,
y
f(x)
the invariant points occur where: y ⴝ 0,
4
label these points A and B; y ⴝ 1, label
√
2
these points C and D; and y ⴝ ⴚ1,
y 3 f(x)
C
D
label these points E and F. Since the
x
A 0
B 3
cube root of a number between
3
E
F
0 and 1 is greater √
than the number,
2
3
the graph of y ⴝ f(x) lies above
4
the graph of y ⴝ f(x) between A and C,
and between B and D. Since the
cube root of a√
number between 0 and ⴚ1 is less than the number, the
graph of y ⴝ 3 f(x) lies below the graph of y ⴝ f(x) between A and E,
and between B and F.
Identify and mark the coordinates of other points.
x
y ⴝ f(x)
ⴚ3
5
0
ⴚ4
3
5
yⴝ
√3
f(x)
1.7
ⴚ1.6
1.7
Join the points with a smooth curve.
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2.1 Properties of Radical Functions—Solutions
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