Name ________________________________________ Date __________________ Class__________________
LESSON
6-6
Reading Strategies
Analyze Information
Looking at the graph of a function, you can determine whether its inverse is
a function. The vertical-line test is used to check whether the graph is a
function. A horizontal-line test is used to check whether the inverse of the
function is also a function. This test allows you to find whether the inverse
is a function without drawing the graph of the inverse function.
Vertical-line Test
If any vertical line intersects the graph at
only one point, then the graph
represents a function.
Horizontal-line Test
If any horizontal line intersects
the graph at only one point, then the
inverse relation is a function.
Solve.
1. The function f (x) = 3 is a linear function. Explain whether the inverse of this
graph is a function.
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2. All linear functions f (x) = mx + b, where m ≠ 0, have an inverse relation that is a function.
Use the horizontal-line test to explain why.
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3.
a. The points (−1, 4) and (5, 4) are on the graph of a function g. What happens
if you draw a horizontal line y = 4 through the graph of the function?
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b. Is the inverse of the function g also a function? Explain why.
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4. Suppose the graph of the inverse of a function is given. Would you use the
horizontal-line test or the vertical-line test to determine whether the graph is a
function? Explain.
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Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
6-50
Holt McDougal Algebra 2
6. y = 2x + 1
3. a. The line y = 4 will intersect the
function at least at two points.
x = 2y + 1
b. No; the function fails the horizontalline test.
x − 1 = 2y
f −1 ( x ) =
x −1
2
4. The vertical-line test should be used to
test whether the graph represents a
function.
domain: all real numbers; range: all real
numbers
6-7 MODELING REAL-WORLD DATA
Challenge
1. Linear functions with a nonzero slope
are always one-to-one.
Practice A
1. a. Yes, the y-values
2. Linear functions with a nonzero slope
are always one-to-one.
b. Second differences
c. Square root function
3. Not one-to-one; fails the horizontal line
test; also try 2 + 1 and 2 − 1 for a
and b.
4. One-to-one
2. Quadratic
3. Linear
4. Exponential
10
5. f(x) = −0.5x2 +
6. f(x) = 0.33x + 2.06
5. Fails horizontal line test, not one-to-one
Practice B
6. Not one-to-one, f(−2) = f(0)
7. Not one-to-one; f(0) = f(1) = 0
1. Exponential
2. Linear
8. One-to-one
3. Quadratic
4. Square root
9. Not one-to-one since (3, 5) is not equal
to (4, 5) but max(3, 5) = 5 = max(4, 5)
5. f(x) = 2x − 9
6. f(x) = −0.4x + 6
7. f(x) = 2.18(1.577)x
8. f(x) = 0.816 x
2
9. f(x) = 0.44x2 + 0.2x + 0.36
Problem Solving
1. a. L(h) = 2.5π h; h ( L ) =
10. f(x) = 1.318x0.378
L
2.5π
11. a. f(x) = 657.3(1.02)x
b. 1634 people
b. h(L) gives the height of a can for a
given lateral surface area.
Practice C
c. 4.5 in.
2. C
3. F
4. B
5. J
1. Quadratic
3. f(x) = 4(0.5)
2.3
Reading Strategies
2. Exponential
x
4. f(x) = −0.8x +
5. f(x) = 2.045x0.336
6. f(x) = 0.88x0.597
7. f(x) = 940(0.15)x
1. The inverse is not a function. The graph
of f is a horizontal line. So the
horizontal line y = 3 intersects the graph
of f at infinitely many points.
8. f(x) = −x2 + 0.35x + 0.8
9. a. f(x) = 2.15(1.1)x
b. 720 billion shares
2. If the slope is not 0, then the graph is a
straight line inclined at an angle. All
horizontal lines will intersect the graph
at only one point, so these graphs will
all have inverse relations that are
functions.
c. 1966
Reteach
1. Linear model
2. Square root model
3. Yes
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A79
Holt McDougal Algebra 2