Name: ___________________________
Precalculus Teacher: _______________
Block: ______ Section 1.5 Combinations of Functions Objective:​
In this lesson you learned how to find arithmetic combinations and compositions of functions. I. Arithmetic Combinations of Functions Just as two real numbers can be combined with arithmetic operations, two functions can be combined by the operations of ___________________________________________________________ to create new functions. The domain of an arithmetic combination of functions ​
f ​
and ​
g consists of . . . Let ​
f ​
and ​
g​
be two functions with overlapping domains. Complete the following arithmetic combinations of ​
f ​
and ​
g​
for all ​
x ​
common to both domains: 1) Sum: (​
f ​
+ ​
g​
)(​
x)​
= ___________________________________ 2) Difference: (​
f ​
­ ​
g​
)(​
x)​
= ___________________________________ 3) Product: 4)Quotient: (​
fg​
)(​
x)​
= ____________________________________ (​
x)​
= _____________________________________ To use a graphing utility to graph the sum of two functions, . . . Example 1:
Let ​
f(​
x)​
​
= 7​
x ​
− 5 and ​
g​
(​
x)​
= 3 − 2​
x ​
. Find (​
f ​
− ​
g​
)(4) Name: ___________________________
Precalculus Teacher: _______________
Block: ______ II. Compositions of Functions The ​
composition​
of the function ​
f ​
with the function ​
g​
, is (​
f ​
⚬ ​
g​
)(​
x)​
, . . . For the composition of the function ​
f ​
with ​
g​
, the domain of f​
⚬ ​
g​
is . . . Example 2:
2​
Let ​
f(​
x)​
​
= 3​
x ​
+ 4 and let ​
g​
(​
x)​
= 2x​
− 1. Find (a)(​
f ​
⚬ ​
g​
)(​
x)​
and (b) (​
g​
⚬ ​
f)​
(​
x)​
. III. Applications of Combinations of Functions The function ​
f(​
x) = 0.06​
x ​
represents the sales tax owed on a purchase with a price tag of ​
x ​
dollars and the function ​
g​
(x) = 0.75​
x ​
represents the sale price of an item with a price tag of ​
x ​
dollars during a 25% off sale. Using one of the combinations of functions discussed in this section, write the function that represents the sales tax owed on an item with a price tag of ​
x ​
dollars during a 25% off sale. IV. Composition of Functions Graphically Find (​
g​
⚬ ​
f)​
(3)
Find (​
f ​
⚬ ​
g​
)(­1)
Name: ___________________________
Precalculus Teacher: _______________
Block: ______ Arithmetic Combinations
1. Given ​
f(​
x)​
​
= 2​
x ​
+ 1 and g(​
x)​
= ​
x2​
​
+ 2​
x ​
− 1 a. Find (​
f ​
+ ​
g​
)(​
x)​
.
b. Evaluate the sum when ​
x ​
= 2.
c. Find (​
f ​
− ​
g​
)(​
x)​
.
d. Evaluate the difference when x = 5.
2. Given ​
f(​
x)​
​
= ​
x2​
​
and ​
g​
(​
x)​
= ​
x ​
− 3 a. Find (​
fg​
)(​
x)​
. b. Evaluate the product when ​
x ​
= 4.
c. Find (​
f/​
g​
)(x).
​
d. Evaluate the quotient when x = 1 Compositions of Functions
3. Given ​
f(​
x)​
​
= ​
x2​
​
+ 2​
x ​
and ​
g​
(​
x)​
= 2​
x ​
+ 1, find the following a. (​
f ​
⚬ ​
g​
)(​
x)​
b. (​
f ​
⚬ ​
g​
)(5)
c. (​
g​
⚬ ​
f)​
(​
x)​
d. (​
g​
⚬ ​
f)​
(5)
Name: ___________________________
Precalculus Teacher: _______________
Block: ______ Composition of Functions Graphically Find ​
(f​
​
⚬​​
g
​)​
(2)
Find (​
g ​
⚬​​
f​
)​
(­3) Finding the Domain of a Composite Function
A. Find the domain of the inside function, if there are restrictions, keep them! B. Construct the new function. If there are additional restrictions, add them to those in Step A. 1. Find the domain: (​
f ​
+ ​
g​
)(​
x)​
f​
(​
x)​
= 1/​
x g(​
x)​
= √​
x 2. Find the domain: (​
f ​
⚬ ​
g​
)(​
x)​
2​
2​
f​
(​
x)​
= ​
x​
− 9 g​
(​
x)​
= √(9 − ​
x​
) 3. Find the domain: (​
f ​
⚬ ​
g​
)(​
x)​
f​
(​
x)​
= 1/x
g​
(​
x)​
= x + 1 4. Find the domain: (​
f ​
⚬ ​
g​
)(​
x)​
2
f​
(​
x)​
= x​
g​
(​
x)​
= √(2 − x) Name: ___________________________
Precalculus Teacher: _______________
Block: ______ Identifying a Composite Function
Write the following functions as a composition of two functions, ​
f(​
x)​
​
and ​
g​
(​
x)​
.
3
1. ​
h​
(​
x)​
= (3​
x ​
− 5)​
2. ​
h​
(​
x)​
=