Download 6-1 Square Root Functions as Inverses

6-1
Square Root Functions as Inverses
TEKS FOCUS
VOCABULARY
ĚSquare root function – A square root
TEKS (2)(C) Describe and analyze the relationship between a function
and its inverse (quadratic and square root, logarithmic and exponential),
including the restriction(s) on domain, which will restrict its range.
function is the inverse of a quadratic
function with a restricted domain.
TEKS (1)(A) Apply mathematics to problems arising in everyday life,
society, and the workplace.
ĚApply – use knowledge or information
for a specific purpose. such as solving a
problem
Additional TEKS (1)(D), (2)(B), (2)(D)
ESSENTIAL UNDERSTANDING
The domain of a quadratic function can be restricted so that its inverse is a square
root function.
Key Concept
Inverse of a Quadratic Function
A horizontal line can intersect the graph of f (x) = x2 in two points—where
f ( -2) = f (2), for example. Thus, a vertical line can intersect the graph of f -1 in
two points. f -1 is not a function because it fails the vertical line test.
y
f(x) x2
⫺4
⫺2
2
O
2
f1(x) x
x
2
⫺2
x
O
y
2
However, you can restrict the domain of f so that the inverse of the restricted function
is a function.
y
y
2
2
f(x) x2 , x ⱖ 0
⫺2
230
Lesson 6-1
O
Square Root Functions as Inverses
x
2
O
f1(x) x
2
4
x
Problem 1
P
TEKS Process Standard (1)(D)
Finding an Inverse
Consider the function f (x) = x2 .
A Write the inverse of the function.
First, rewrite the equation as y = x2 . Since the graph of an inverse is a reflection
across the line y = x, you can switch x and y in the equation to get an equation
for the inverse.
y = x2
x = y2
{2x = y
Switch x and y.
Find the square root of each side to solve for y.
Using f -1 notation, this equation can also be written as f -1(x) = {2x.
B Analyze and describe the relationship between the function and its inverse,
including restrictions on domain and range.
What will the graph
of the inverse look
like?
Imagine folding the graph
of y = x 2 along the
line y = x. You will get
a parabola in the same
shape as y = x 2 , but
opening to the right.
The function and its inverse represent inverse operations.
The domain of y = x2 is all real numbers. Since x2 must be nonnegative, the
range is y Ú 0. Square root functions are defined for nonnegative real numbers,
so the domain of y = {2x is x Ú 0, and the range is all real numbers.
C Graph the inverse.
To graph y = {2x, first graph y = 2x, then graph y = - 2x. The graphs of f (x)
and f -1(x) are parabolas, each a reflection of the other across the line y = x. The
graph of f -1(x) shows that it is not a function.
y
2
x
O
-2
5
y=± x
D Restrict the domain of f so that the inverse of f (x) = x2 is a function. What is
the equation of the inverse?
When the domain of f (x) = x2 is restricted to x Ú 0, the range is y Ú 0. The
inverse of the restricted function is f -1(x) = 2x with a domain of x Ú 0. The
range is y Ú 0. The graph shows that the inverse is a function because it passes
the vertical line test.
y
y
2
2
f(x) x2 , x ⱖ 0
⫺2
O
x
2
O
f1(x) x
2
4
x
PearsonTEXAS.com
231
Problem 2
P
Graphing a Function and Its Inverse
Consider the function f (x) = x2 + 3 with domain x # 0, and its inverse,
f −1(x) = 2x − 3.
How are the domain
and range of a
function and its
inverse related?
The domain of a function
is the range of its inverse,
and the range of a
function is the domain
of its inverse.
A Analyze and describe the relationship between the function and its inverse.
Since f (x) is a quadratic function with an appropriately restricted domain, the inverse
is a square root function. The graph of f -1(x) is a reflection of the graph of f (x) across
the line y = x.
B What are the domain and range of f (x) and f −1(x)? Write your answers as
inequalities.
The domain of f is restricted to x Ú 0. The range
of f is y Ú 3.
The domain of f -1 is x Ú 3, and the range of
f -1 is y Ú 0.
C Graph f (x) and f −1(x).
+ 3, translate the graph of
To graph f (x) =
f (x) = x2 up three units and only graph x-values
greater than or equal to zero. To graph the inverse,
reflect the graph of f (x) across the line y = x.
x2
Problem
bl
3
f(x)
8
6
4
2
O
TEKS Process Standard (1)(A)
The function d = 4.9t 2 represents the distance d,
in meters, that an object falls in t seconds due to
Earth’s gravity. Find the inverse of this function.
How long, in seconds, does it take for the cliff diver
shown to reach the water below?
d = 4.9t 2
d
t 2 = 4.9
232
Lesson 6-1
x
5
f−1(x) = x− 3, x ≥ 3
-2
Finding the Inverse of a Formula
Why shouldn’t you
interchange the
variables?
Interchanging the
variables leads to a false
relationship between
distance and time.
f(x) = x2 + 3,
x≥0
d
t = 5 4.9
24
= 5 4.9
≈ 2.2
Solve for t.
Do not switch the variables.
Time must be nonnegative.
Substitute 24 for d.
Use a calculator.
It
I will take about 2.2 seconds for the diver to reach
the
t water.
Square Root Functions as Inverses
24 meters
Problem 4
P
Composing Functions
How does
composition show
that two functions
are inverses?
The composition of a
function and its inverse
is the identity function.
So as long as you choose
an x value within the
domain, if the functions
are inverses, the result of
the composition should
be the original x value.
U composition to show that f (x) = x2 + 7 with domain x # 0 and
Use
g(x)
= 1x − 7 are inverse functions.
g
In
I general, if ( g ∘ f )(x) = x and ( f ∘ g)(x) = x for x in the domains of f and g,
respectively, then
f and g are inverse functions.
r
Check:
C
( f ∘ g)(x) = f (g(x))
(g ∘ f )(x) = g( f (x))
(
= f 2x - 7
(
= 2x - 7
)
)2 + 7
= g(x2 + 7)
= 2(x2 + 7) - 7
=x-7+7
= 2x2
=x
= x = x (since x Ú 0)
NLINE
HO
ME
RK
O
So f (x) and g(x) are inverse functions.
S
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
For Exercises 1–2, write the inverse of each function.
1. Write the inverse of f (x) = 5x - 4.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
2. Write the inverse of f (x) = (x + 8)2 . Give the domain and range of the function
and its inverse.
3. Write and graph the inverse of y = - x2 + 2.
4. Use Multiple Representations to Communicate Mathematical Ideas (1)(D)
Find the inverse of y = 4x2 . Graph both the function and its inverse. Explain
how the equations and graphs show the relationship between the function and
its inverse.
5. Graph the function f (x) = 13 x - 2 and its inverse, f -1(x) = 3x + 6.
6. Graph the function f (x) = (x + 2)2 and its inverse, f -1(x) = { 2x - 2. How
would you restrict the domain of f so that its inverse is a function? What is the
equation of the inverse function?
7. a. What are the domain and range of f (x) = - x2 + 4 and its inverse,
f -1(x) = {24 - x? Write your answers in interval notation.
b. Restrict the domain of f so that its inverse is a function. What is the equation
of the inverse function?
8. Analyze Mathematical Relationships (1)(F) The area of a circle is given by
the equation A = pr 2 , where r is the radius of the circle. The inverse function is
A
r = 5p
. Analyze and describe the relationship between the functions. Write
the domain and range of both functions as inequalities.
PearsonTEXAS.com
233
Use composition to show that f and g are inverse functions.
9. f (x) = 12 x + 3 and g(x) = 2x - 6
x
10. f (x) = 5 5 and g(x) = 5x2, x 7 0
1-x
11. f (x) = 1 - 2x2, x Ú 0 and g(x) = 5 2
12. Nina belongs to a gym that charges $35 per month plus a $95 enrollment fee.
She has found that the equation f (x) = 35x + 95 gives the total amount she
has paid for x months. Find the inverse of this function. Then use composition
to check your answer.
13. The formula for converting from Celsius to Fahrenheit temperatures
is F = 95C + 32.
a. Find the inverse of the formula. Is the inverse a function?
b. Use the inverse to find the Celsius temperature that corresponds to 25°F.
14. V = 43pr 3 is the formula for the volume of a sphere.
a. Find the inverse of the formula. Is the inverse a function?
b. Use the inverse to find the radius of a sphere that has a volume of 35,000 ft3.
15. Apply Mathematics (1)(A) The velocity of the water that flows from an
opening at the base of a tank depends on the height of water above the
opening. The function v (x) = 12gx models the velocity v in feet per second
where g, the acceleration due to gravity, is about 32 ft>s2 and x is the height in
feet of the water. What is the depth of water when the flow is 40 ft/s, and when
the flow is 20 ft/s?
16. Let f (x) = 3x2 - 4 and g (x) = x - 2. Calculate ( f ∘ g -1)(x) for x = -3.
17. Explain Mathematical Ideas (1)(G) Explain how you can find the range of the
inverse of f (x) = 1x - 1 without finding the inverse itself.
For each function, find the inverse and the domain and range of the
function and its inverse. Determine whether the inverse is a function.
18. f (x) = - 1x
19. f (x) = 1x + 3
20. f (x) = 1-x + 3
21. f (x) = 1x + 2
x2
23. f (x) = 12
22. f (x) = 2
24. f (x) = (x 26. f (x) =
234
Lesson 6-1
x
4)2
1
(x + 1)2
Square Root Functions as Inverses
25. f (x) = (7 - x)2
27. f (x) = 4 - 21x
28. a. Display Mathematical Ideas (1)(G) Copy the mapping diagram
at the right. Complete it by writing members of the domain and
range and connecting them with arrows so that r is a function and
r -1 is not a function.
Relation r
Domain
Range
b. Repeat part (a) so that r is not a function and r -1 is a function.
29. Explain Mathematical Ideas (1)(G) Relation r has one element in its
domain and two elements in its range. Is r a function? Is the inverse of r
a function? Explain.
30. Apply Mathematics (1)(A) Write a function that gives the length of the
hypotenuse of an isosceles right triangle with side length s. Evaluate the
inverse of the function to find the side length of an isosceles right triangle
with a hypotenuse of 6 in.
3
31. For the function f (x) = 1
2x, find f -1(x). Then determine the value of
x when f (x) = 16.
TEXAS Test Practice
T
32. Which pair of words makes this sentence FALSE?
The product of two ____(I)____ numbers is always a(n) ____(II)____ number.
A. (I) complex; (II) complex
B. (I) real; (II) complex
C. (I) rational; (II) real
D. (I) imaginary; (II) imaginary
33. If f (x) = x + 1 and g (x) = x2 - 3x - 4, what is ( f ∘ g)(x)?
F. x2 - 3x - 3
G. x2 - x - 6
H. x2 - x
J. x2 - x - 3
2 3
( )
2
34. What is the simplified form of a3b4 ?
4
9
A. a9b16
4 3
B. a3b2
C. ab
17
D. (ab) 6
35. Let f (x) = (x + 1)2 - 2. Find the x- and y-intercepts of f (x) and the inverse of
f (x). Is the inverse a function?
PearsonTEXAS.com
235