Download 6-1 Evaluate nth Roots and Use Rational Exponents

6-3 Perform Function Operations and Composition
Name__________________
Objective: To perform operations with functions.
Algebra 2 Standards 24.0 and 25.0
*Operations on Functions: Let f and g be any two functions. A new function h can be defined by
performing any of the four basic operations on f and g.
Operation
Addition
Definition
h(x) = f(x) + g (x)
Subtraction
Multiplication
Division
h(x) = f(x) – g (x)
h(x) = f(x) · g (x)
h(x) =
Example: f(x) = 5x, g(x) = x + 2
f ( x)
g ( x)
The domain of h consists of the x-values that are in the domains of both f and g.
The domain of the quotient does not include x-values for which g(x) = 0.
*Power Function: y  axb where a is a real number and b is a rational number.
Example 1: Let f ( x)  5 x1/3 and g ( x)  11x1/3 . Find the following.
a. f(x) + g(x)
b. f(x) – g(x)
c. the domains of parts a and b
You Try: Let f ( x)  2 x 2/3 and g ( x)  7 x 2/3 . Find the following.
a. f(x) + g(x)
b. f(x) – g(x)
c. the domains of parts a and b
Example 2: Let f( x) = 8x and g(x) = 2x5/6. Find the following.
f ( x)
a. f(x) · g (x)
b.
g ( x)
You Try: Let f( x) = 3x and g(x) = x1/5. Find the following.
f ( x)
a. f(x) · g (x)
b.
g ( x)
Algebra 2 Ch.6B Notes-page1
Example 3: A small company sells computer printers over the Internet. The company’s total monthly
revenue (R) and costs (C) are modeled by the functions R(x) = 120x and C(x) = 2500 + 75x where x is the
number of printers sold.
a. Find R(x) – C(x)
b. Explain what this difference means.
*Composition of Functions:
The composition of a function g
with a function f is:
Domain of f
h(x) = g(f(x))
Range of f
The domain of h is the set of all
x-values such that x is in the domain
of f and f(x) is in the domain of g.
Domain of g
Example 4: Let f(x) = 3x – 4 and g(x) = x2 – 1. What is the value of f(g(-3))?
A.
B.
C.
D.
-34
8
20
168
You Try: Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.
a. g(f(5))
b. f(g(5))
c. f(f(5))
d. g(g(5))
Algebra 2 Ch.6B Notes-page2
Range of g .
Example 5: Your starting wage for your part-time job was $6 an hour. All employees get a 5% raise after
6 months. You are given an additional raise of $.75 per hour as a reward for your outstanding work. Find
your new hourly wage if the 5% raise is applied before the $.75 raise.
You Try: Use example 5 and find your new hourly wage if the $.75 raise is applied before the 5% raise.
Example 6: Let f(x) = 6x 2 and g(x) = 4x + 5. Find the following.
a. f(g(x))
b. g(f(x))
c. g(g(x))
You Try: Let f(x) = 2x 1 and g(x) = 2x + 7. Find f(g(x)), g(f(x)) and f(f(x)).
Algebra 2 Ch.6B Notes-page3
6-4 Use Inverse Functions
Name:__________________
Objective: To find inverse functions.
Algebra 2 Standards 24.0 and 25.0
*Recall that an inverse relation just switches the x and y values (i.e., your domain becomes your range
and your range becomes the domain). The graph of an inverse relation is a
______________________________ of the original graph.
*To find an inverse relation, switch x and y and then solve for y.
Example 1: Find the equation for the inverse of the relation y = 4x + 2.
*Inverse Functions: Functions f and g are inverses of each other provided:
f(g(x)) = x
and
g(f(x)) = x
1
The function g is denoted by f and read “f inverse.”
Example 2: Verify that f(x) = 4x + 2 and f 1 ( x) 
1
1
x
4
2
Example 3: Find the inverse of f(x) = 2x – 1 Then verify that your result and the original function are
inverses.
You Try: Find the inverse of the given function. Then verify that your result and the original function are
inverses.
a. f(x) = x + 4
Algebra 2 Ch.6B Notes-page4
b. f(x) = -3x + 1
*When finding inverses of models (real life examples), do not switch the variables because the
letters represent the specific quantities. Just solve for the other variable.
Example 4: A small company produces greeting cards. The cost C (in dollars) of producing n greeting
cards per month can be modeled by the function C = 360 + 0.60n.
a. Find the inverse of the model.
b. Use the inverse function to find the number of greeting cards produced in a month in which the
company’s total cost to produce the cards was $615.
*Inverses of Nonlinear Functions: Copy the original graph and find the reflection in the line y = x.
Notice that the inverse of g(x) = x3 is a function, but that the inverse of f(x) = x2 is not a function.
f(x) = x2
g(x) = x3
*If the domain of f(x) = x2 is restricted to only nonnegative real numbers,
then the inverse is a function.
Example 5: Find the inverse of f(x) = x2 + 2, x < 0. Then graph f and f 1 .
Algebra 2 Ch.6B Notes-page5
*You can use the graph of a function f to determine whether the inverse of f is a function
by applying the ________________________ line test. The inverse of a function f is also a function
if and only if no horizontal line intersects the graph of f more than once.
Example 6: Consider the function f(x) = 3x5 – 2. Determine whether the inverse of f is a function. Then
find the inverse.
You Try: Find the inverse of the function. Then graph the function and its inverse.
a. f(x) =  x 3  4
b. f(x) = 2x5 + 3
Example 7: The population of a town can be modeled by P  16,500t 0.15 , where t is the number of years since
1998. Find the inverse model that gives the number of years as a function on the population.
Algebra 2 Ch.6B Notes-page6
6-5 Graph Square Root and Cube root Functions
Name:__________________
Objective: To find inverse functions.
Algebra 2 Standards 24.0 and 25.0
1) Square Root function
f ( x)  x
 Domain is x  0
 Range is y  0
2) Cube root Function
g ( x)  3 x
Domain and Range are
all real numbers.
3) Transformation of Radical Functions
f ( x)  x
y a xh k


a  changes steepness :
f ( x)  3 x

y  a3 x  h  k
a  1 stretches f(x) vertically.
0  a  1 shrinks f(x) vertically.
negative a reflects f(x) over the x-axis.


h  shifts left or right:
k  shifts up or down
Example 1: Graph the function and state the domain and range.
a. y  2 x
Algebra 2 Ch.6B Notes-page7
b. y  
13
x
2
You Try: Graph the function and state the domain and range.
a. y  2 3 x
b. y  2 x
Example 2: Graph y  3 x  3  2 and state the domain and range.
Example 3: Graph y  2 3 x  3  2 and state the domain and range.
Algebra 2 Ch.6B Notes-page8
6-6 Solve Radical Equations
Name: _______________________
Objective: To solve radical equations.
Algebra 2 Standard 12.1
*Equations that have variables inside radicals are called radical equations.
*How to solve radical equations:
1. Isolate the radical on one side of the equation.
2. Raise each side of the equation to the same power to eliminate the radical.
3. Solve the equation.
4. Check your solution.
Ex. 1: Solve
a. 5 x  9  11
b.
You Try: Solve
a. x  25  4
b. 2 3 x  3  4
3
x  9  1
Ex. 2: If an 8-foot ladder is leaned against a wall, the height is given by h  x   64  x2 ,
where x is the distance along the floor from the base of the ladder to the wall. Estimate the distance that
the ladder should be placed from the wall in order for the ladder to reach 7 feet off the floor.
Algebra 2 Ch.6B Notes-page9
Ex. 3: Solve 7 x3/5  56
A.
B.
C.
D.
4.8
8
12.8
32
Ex. 4: Solve  x  4 
2/3
 9  16
You Try:
 x  2
1/3
3 7
*Raising each side of an equation to the same power sometimes creates extraneous solutions.
This is a solution that appears to be a solution, but when substituted back into the original
equation, does not actually work. Be sure to check all apparent solutions when you have
variables more than one places.
Ex. 5: Solve
x 6  x 8
You Try: Solve
2x  5  x  7
Algebra 2 Ch.6B Notes-page10
Ex. 6. Solve 10 x  9  x  3