1. Algebra and Composition of Functions
Section 11.1
Algebra and Composition of Functions

2. Objectives Add, subtract, multiply, and divide functions
Find the composition of functions
Use graphs to evaluate functions
Use composite functions to solve problems

3. Objective 1: Add, Subtract, Multiply, and Divide Functions
Operations on Functions: If the domains and ranges of functions ƒ and g are subsets of the real numbers, then:
Sum:
(ƒ + g)(x) = ƒ(x) + g(x)
Difference:
(ƒ – g)(x) = ƒ(x) – g(x)
Product:
(ƒ · g)(x) = ƒ(x) · g(x)
Quotient:
The domain of each sum, difference, and product function shown above is the set of real numbers x that are in the domains of both ƒ and g. The domain of the quotient function is the set of real numbers x that are the domains of ƒ and g, excluding any values of x where g(x) = 0.
Read as “f plus g of x equals f of x plus g of x.”
Read as “f minus g of x equals f of x minus g of x.”
Read as “f times g of x equals f of x times g of x.”
Read as “f divided by g of x equals f of x divided by g of x.”

4. Objective 1: Add, Subtract, Multiply, and Divide Functions
There is a relationship that can be seen between the graphs of two functions and the graph of their sum (or difference) function.
For example, in the illustration, the graph of ƒ + g, which gives the total number of elementary and secondary students in the United States, can be found by adding the graph of ƒ, which gives the number of elementary students, to the graph of g, which gives the number of secondary students. For any given x-value, we simply add the two corresponding y-values to get the graph of the sum function ƒ + g.

5. EXAMPLE 1
Let ƒ(x) = 2x2 + 1 and g(x) = 5x – 3. Find each function and its domain: a. ƒ + g b. ƒ – g
c. ƒ · g d. ƒ / g
Strategy We will add, subtract, multiply, and divide the functions as if they were binomials..
Why We add because of the plus symbol in ƒ + g, and we subtract because of the minus symbol in ƒ – g.

6. EXAMPLE 1
Let ƒ(x) = 2x2 + 1 and g(x) = 5x – 3. Find each function and its domain: a. ƒ + g b. ƒ – g
c. ƒ · g d. ƒ / g
Solution
The domain of ƒ + g is the set of real numbers that are in the domain of both ƒ and g. Since the domain of both ƒ and g is the interval (–∞, ∞) the domain of ƒ + g is the interval (–∞, ∞).

7. EXAMPLE 1
Let ƒ(x) = 2x2 + 1 and g(x) = 5x – 3. Find each function and its domain: a. ƒ + g b. ƒ – g
c. ƒ · g d. ƒ / g
Solution
Since the domain of both ƒ and g is (–∞, ∞) the domain of ƒ – g is the interval (–∞, ∞).

8. EXAMPLE 1
Let ƒ(x) = 2x2 + 1 and g(x) = 5x – 3. Find each function and its domain: a. ƒ + g b. ƒ – g
c. ƒ · g d. ƒ / g
Solution
The domain of ƒ · g is the set of real numbers that are in the domain of both ƒ and g. Since the domain of both ƒ and g is (–∞, ∞), the domain of ƒ · g is (–∞, ∞). 8

9. EXAMPLE 1
Let ƒ(x) = 2x2 + 1 and g(x) = 5x – 3. Find each function and its domain: a. ƒ + g b. ƒ – g
c. ƒ · g d. ƒ / g
Solution
Since the denominator of the fraction cannot be 0, it follows that this function is undefined if 5x – 3 = 0. If we solve for x, we see that x cannot be . Thus, the domain of ƒ / g is the union of two intervals: 9

10. Objective 2: Find the Composition of Functions
Often one quantity is a function of a second quantity that depends, in turn, on a third quantity. For example, the cost of a car trip is a function of the gasoline consumed. The amount of gasoline consumed, in turn, is a function of the number of miles driven. Such chains of dependence can be analyzed mathematically as compositions of functions.
Suppose that y = ƒ(x) and y = g (x) define two functions. Any number x in the domain of g will produce the corresponding value g(x) in the range of g. If g(x) is in the domain of function ƒ, then g(x) can be substituted into ƒ, and a corresponding value ƒ(g(x)) will be determined. This two-step process defines a new function, called a composite function, denoted by ƒ ○ g. (This is read as “ƒ composed with g” or “the composition of ƒ and g” or “ƒ circle g.”)

11. Objective 2: Find the Composition of Functions
The function machines illustrate the composition ƒ ○ g. When we put a number into the function g a value g(x) comes out.
The value g (x) then goes into function ƒ, which transforms g(x) into ƒ(g(x)). (This is read as “ƒ of g of x.”) If the function machines for g and ƒ were connected to make a single machine, that machine would be named ƒ ○ g.

12. Objective 2: Find the Composition of Functions
The composite function ƒ ○ g is defined by
(ƒ ○ g )(x) = ƒ(g(x))
To be in the domain of the composite function ƒ ○ g, a number x has to be in the domain of g and the output of g must be in the domain of ƒ. Thus, the domain of ƒ ○ g consists of those numbers x that are in the domain of g, and for which g(x) is in the domain of ƒ.

13. EXAMPLE 3
Strategy In part (a), we will find ƒ(g(9)). In part (b), we will find ƒ(g(x)). In part (c), we will find g(ƒ(–2)).
Why To evaluate a composition function written with the circle ○ notation, we rewrite it using nested parentheses: (ƒ ○ g)(x) = ƒ(g(x)).

14. EXAMPLE 3
Solution a.
(ƒ ○ g)(9) means ƒ(g(9)). In figure (a), function g receives the number 9, subtracts 4, and releases the number g(9) = 5. Then 5 goes into the ƒ function, which doubles 5 and adds 1. The final result, 11, is the output of the composite function
(ƒ ○ g).

15. EXAMPLE 3
Solution b.
Distribute the multiplication by 2.
Combine like terms.
(ƒ ○ g)(x) means ƒ(g(x)). In figure (a), function g receives the number x, subtracts 4, and releases the number x – 4. Then x – 4 goes into the ƒ function, which doubles x – 4 and adds 1. The final result, 2x – 7, is the output of the composite function (ƒ ○ g).

16. EXAMPLE 3
Solution c.
Do the subtraction.
(g ○ ƒ)(–2) means g(ƒ(–2)). In figure (b), function ƒ receives the number –2, doubles it and adds 1, and releases –3 into the g function. Function g subtracts 4 from –3 and outputs a final result of –7.

17. Objective 3: Use Graphs to Evaluate Functions

18. EXAMPLE 4
Refer to the graphs of functions ƒ and g to find each of the following.
a. (ƒ + g) (–4)
b. (ƒ · g)(2)
c. (ƒ ○ g)(–3)
Strategy We will express the sum, product, and composition functions in terms of the functions from which they are formed.
Why We can evaluate sum, product, and composition functions at a given x-value by evaluating each function from which they are formed at that x-value.

19. EXAMPLE 4
Refer to the graphs of functions ƒ and g to find each of the following.
a. (ƒ + g) (–4)
b. (ƒ · g)(2)
c. (ƒ ○ g)(–3)
Solution

20. EXAMPLE 4
Refer to the graphs of functions ƒ and g to find each of the following.
a. (ƒ + g) (–4)
b. (ƒ · g)(2)
c. (ƒ ○ g)(–3)
Solution

21. EXAMPLE 4
Refer to the graphs of functions ƒ and g to find each of the following.
a. (ƒ + g) (–4)
b. (ƒ · g)(2)
c. (ƒ ○ g)(–3)
Solution

22. Objective 4: Use Composite Functions to Solve Problems

23. EXAMPLE 6
Biological Research. A specimen is stored in refrigeration at a temperature of 15° Fahrenheit. Biologists remove the specimen and warm it at a controlled rate of 3°F per hour. Express its Celsius temperature as a function of the time since it was removed from refrigeration.
Strategy We will express the Fahrenheit temperature of the specimen as a function of the time t since it was removed from refrigeration. Then we will express the Celsius temperature of the specimen as a function of its Fahrenheit temperature and find the composition of the two functions.
Why The Celsius temperature of the specimen is a function of its Fahrenheit temperature. Its Fahrenheit temperature is a function of the time since it was removed from refrigeration. This chain of dependence suggests that we write a composition of functions.

24. EXAMPLE 6
Biological Research. A specimen is stored in refrigeration at a temperature of 15° Fahrenheit. Biologists remove the specimen and warm it at a controlled rate of 3°F per hour. Express its Celsius temperature as a function of the time since it was removed from refrigeration.
Solution
The temperature of the specimen is 15°F when the time t = 0. Because it warms at a rate of 3°F per hour, its initial temperature of 15°F increases by 3t °F in t hours. The Fahrenheit temperature at time t of the specimen is given by the function
The Celsius temperature C is a function of this Fahrenheit temperature F, given by the function

25. EXAMPLE 6
Biological Research. A specimen is stored in refrigeration at a temperature of 15° Fahrenheit. Biologists remove the specimen and warm it at a controlled rate of 3°F per hour. Express its Celsius temperature as a function of the time since it was removed from refrigeration.
Solution
To express the specimen’s Celsius temperature as a function of time, we find the composite function (C ○ F)(t).
The composite function, , gives the temperature of the specimen in degrees Celsius t hours after it is removed from refrigeration.