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Checkpoint: Assess Your Understanding, pages 124–128
2.1
1. Multiple Choice Given the graph of the function y ⫽ f(x), which
graph below right represents y =
√
f(x)?
y
4
2
D
0
2
2
y f(x)
4
©P DO NOT COPY.
A
2
x
2
y
x
4
2
0
C
2
2
B
Chapter 2: Radical and Rational Functions—Checkpoint—Solutions
17
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2. For each function y ⫽ f(x) graphed below:
√
• Sketch the graph of y = f(x).
√
• State the domain and range of y = f(x).
• Explain why the domains are different and the ranges are
different.
a)
b)
y
4
y f (x )
√
y f(x)
x
6
4
2
4
0
y
y f(x)
2
2
y
x
0
2
4
√
f (x )
6
4
Mark points where y ⴝ 0 or y ⴝ 1.
√
The graph of y ⴝ f(x) is above
the graph of y ⴝ f(x) between
these points. Choose, then mark
other points on the graph of
√
y ⴝ f(x) .
√
x
y ⴝ f(x)
ⴚ3
2
1.4
0
4
2
yⴝ
f(x)
Join the points with a smooth
curve.
Domain is: x » ⴚ6
Range is: y » 0
√
y ⴝ f(x) is not defined
for ⴚ2<x<6. Mark points
where y ⴝ 0 or y ⴝ 1.
Choose, then mark other
points on the graph of
√
y ⴝ f(x).
√
f(x)
x
y ⴝ f(x)
ⴚ4
5
2.2
8
5
2.2
yⴝ
Join the points with 2 smooth
curves.
Domain is: x ◊ ⴚ2 or x » 6
Range is: y » 0
The domain of a linear function or a quadratic function is all real values
of x, but the square root of a negative number is undefined,
so any value of x that makes the radicand negative is not in the
domain of a radical function.
The range of the linear function is all real values of y, and the range
of the quadratic function is all real values of y that are greater than
or equal to ⴚ4. The principal square root of a number is always 0 or
positive, so the range of the radical functions is restricted to these
values of y.
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Chapter 2: Radical and Rational Functions—Checkpoint—Solutions
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3. Solve each radical equation by graphing. Give the solution to the
nearest tenth.
a) -x + 3 =
√
2x - 1
b)
Write the equation as:
√
ⴚx ⴙ 3 ⴚ 2x ⴚ 1 ⴝ 0
Graph the related function:
√
f(x) ⴝ ⴚx ⴙ 3 ⴚ 2x ⴚ 1
Use graphing technology to
determine the approximate
zero: 1.5505103
So, the solution is: x 1.6
√
x + 2 = 5 -
√
3x + 4
Write the equation as:
√
√
x ⴙ 2 ⴚ 5 ⴙ 3x ⴙ 4 ⴝ 0
Graph the related function:
√
√
f(x) ⴝ x ⴙ 2 ⴚ 5 ⴙ 3x ⴙ 4
Use graphing technology to determine
the approximate zero: 1.779514
So, the solution is: x 1.8
2.2
4. Use graphing technology to graph each rational function.
Identify any non-permissible values of x and the equations of any
horizontal asymptotes.
a) y =
3x
x + 4
b) y =
Since x ⴙ 4 ⴝ 0, then x ⴝ ⴚ4
The vertical asymptote has
equation x ⴝ ⴚ4.
The horizontal asymptote has
equation y ⴝ 3.
c) y =
x2 - 4
3x
3x
x - 4
2
Since x2 ⴚ 4 ⴝ 0, then x ⴝ —2
The vertical asymptotes have
equations x ⴝ 2 and x ⴝ ⴚ2.
The horizontal asymptote has
equation y ⴝ 0.
d) y =
Since 3x ⴝ 0, then x ⴝ 0
The vertical asymptote has
equation x ⴝ 0.
There is no horizontal asymptote.
x2 - 4x
3x
Since 3x ⴝ 0, then x ⴝ 0
There is a hole at x ⴝ 0.
There is no horizontal asymptote.
2.3
5. Multiple Choice Which function has a graph with a hole?
x + 4
x - 4
A. y = 2
B. y = 2
2x + 8x
2x + 8x
C. y =
4x + 4
2x2 + 8x
©P DO NOT COPY.
D. y =
x + 4
2x2 - 8x
Chapter 2: Radical and Rational Functions—Checkpoint—Solutions
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6. Match each function to its graph. Justify your choice.
i) Graph A
ii) Graph B
y
4
40
y f (x)
y
y g(x)
20
2
x
x
4
2
0
2
0
4
2
4
6
20
2
40
iii) Graph C
iv) Graph D
y
y
x
0
2
4
10
x
8
12
2
y h(x)
- 12x
a) y = x - 3
There is a vertical asymptote
with equation x ⴝ 3. The
degrees of the numerator and
denominator are equal, so
there is a horizontal asymptote
that is not the x-axis. The
function matches Graph B.
x2
c) y = x - 3
There is a vertical asymptote
with equation x ⴝ 3.
The degree of the numerator
is 1 more than the degree of
the denominator, so there is
an oblique asymptote.
The function matches Graph D.
20
y k (x)
20
4
0
4
6
10
b) y =
2x2 - x - 15
3 - x
(2x ⴙ 5)(x ⴚ 3)
,
3ⴚx
ⴚ (2x ⴙ 5)(x ⴚ 3)
or y ⴝ
xⴚ3
Factor: y ⴝ
There is a hole at x ⴝ 3. The
function matches Graph C.
d) y =
x2
x - 3
2
The function is not defined for
√
x2 ⴚ 3 ⴝ 0; that is, x ⴝ — 3.
So, there are vertical
√
asymptotes at x ⴝ ⴚ 3 and
√
x ⴝ 3. The degrees of the
numerator and denominator
are equal, so there is a
horizontal asymptote that is
not the x-axis. The function
matches Graph A.
Chapter 2: Radical and Rational Functions—Checkpoint—Solutions
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7. For the graph of each rational function below, determine without
technology:
i) the equations of any asymptotes and the coordinates of any hole
ii) the domain of the function
Use graphing technology to verify the characteristics.
a) y =
2x2
25 - x2
i) The function is undefined when 25 ⴚ x2 ⴝ 0; that is,
when x ⴝ —5. There are no common factors, so there are vertical
asymptotes with equations x ⴝ 5 and x ⴝ ⴚ5.
The degrees of the numerator and denominator are equal, so there is
a horizontal asymptote. The leading coefficients of the numerator and
denominator are 2 and ⴚ1, respectively.
So, the horizontal asymptote has equation: y ⴝ ⴚ2
ii) The domain is: x ⴝ —5
b) y =
- 2x2 - 6x
x + 3
i) The function is undefined when x ⴙ 3 ⴝ 0; that is, when x ⴝ ⴚ3.
ⴚ 2x(x ⴙ 3)
Factor: y ⴝ
xⴙ3
There is a hole at x ⴝ ⴚ3. The function is: y ⴝ ⴚ2x, x ⴝ ⴚ3
The coordinates of the hole are: (ⴚ3, 6)
ii) The domain is: x ⴝ ⴚ3
8. Solve each rational equation by graphing. Give the solution to the
nearest tenth.
3x - 5
a) x - 2 = x - 3
b)
x2 + 3x - 5
= -5
x - 1
Graph a related function:
Graph a related function:
3x ⴚ 5
f(x) ⴝ x ⴚ 2 ⴚ
xⴚ3
f(x) ⴝ
Use graphing technology to
determine the zeros:
x 1.8 or x 6.2
Use graphing technology to
determine the zeros:
x ⬟ ⴚ9.1 or x ⬟ 1.1
©P DO NOT COPY.
x2 ⴙ 3x ⴚ 5
ⴙ5
xⴚ1
Chapter 2: Radical and Rational Functions—Checkpoint—Solutions
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