3. Functions and Graphs

Form Composite Functions

Combinations of Functions; Composite Functions

Learn How to Compose Two Rational Functions and Simplify

Express A Function As A Composition Of Two Functions

In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = |2x-5|

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = ∛(x² – 9)

In Exercises 1–30, find the domain of each function. h(x) = √(x −2)+ √(x +3)

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1-x^2

In Exercises 67-74, find
a. (fog) (x)
b. the domain of f o g. 
f(x) = x/(x+1), g(x) = 4/x

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/cb8fec2cd9c2a025-1674042375567.jpg" alt="" /> 
Find (fg) (2).

In Exercises 31–50, find f/g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 18

In Exercises 31–50, find fg and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=8x+12, g(x)=3x-1

Use the table to evaluate each expression, if possible.  (f-g)(3)

In Exercises 31–50, find fg and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 16

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ-g)(4)

In Exercises 31–50, find fg and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

In Exercises 51–66, find 
a. (fog) (2)
b. (go f) (2)
f(x) = x²+2, g(x) = x² – 2

Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2
(ƒ/g)(2)

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2). 
f(x) = 1/x, g(x)= 1/x

For the pair of functions defined, find (ƒ+g)(x). Give the domain of each. See Example 2. ƒ(x)=√(4x-1), g(x)=1/x

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Solve and check: (x-1)/5 - (x+3)/2 = 1- x/4

Manipulating Functions Algebraically and Evaluating Composite Functions

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x) 
f(x)=4x-3, g(x) = 5x² - 2

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=8x+12, g(x)=3x-1

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

Decomposing Functions - Composition of Functions

In Exercises 31–50, find f/g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

In Exercises 31–50, find fg and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

Let ƒ(x) = 3x^2 - 4 and g(x) = x^2 - 3x -4. Find each of the following. (f+g)(2k)

In Exercises 1–30, find the domain of each function. g(x) = 2/(x+5)

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=-6x+9, g(x)=5x+7

Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (g ○ ƒ)(3)

Let ƒ(x) = 3x^2 - 4 and g(x) = x^2 - 3x -4. Find each of the following. (f/g)(-1)

Combining functions by adding and subtracting

In Exercises 59-64, let
f(x) = 2x - 5 
g(x) = 4x - 1
h(x) = x² + x + 2.
Evaluate the indicated function without finding an equation for the function. g (f[h (1)])

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x^2

In Exercises 51–66, find 
a. (fog) (2)
b. (go f) (2) 
f(x)=4x-3, g(x) = 5x² - 2

Understand restrictions on functions in terms of domain and range

In Exercises 1–30, find the domain of each function. f(x) = √(x - 3)

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = √x, g(x) = x − 4

Combining functions by dividing

In Exercises 31–50, find fg and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = √x, g(x) = x − 4

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=-x^2

In Exercises 31–50, find f/g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Finding Domain of Functions Involving Radicals (Square Roots to be More Precise!) - Example 2

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=-2x+5

In Exercises 1–30, find the domain of each function. f(x) = √(24 - 2x)

Domain of a Composition of Functions, Example 2

Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. the domain of ƒ ○ g

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/ca2974f939c2b767-1674043290958.jpg" alt="" />
Find the domain of ƒ + g.

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x+2, g(x)=x^4+x^2-4

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x).
f(x) = √x, g(x) = x − 1

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = x+4, g(x) = 2x + 1

In Exercises 87–88, find a. (f ○ g)(x); b. the domain of (f ○ g). f(x) = (x + 1)/(x - 2), g(x) = 1/x

In Exercises 67-74, find
a. (fog) (x)
b. the domain of f o g. 
f(x) = x² + 4, g(x) = √(1 − x)

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒg)(-3)

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x^2+3x+1

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1/x^2

For the pair of functions defined, find (f/g)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-8

In Exercises 1–30, find the domain of each function. f(x)=3(x-4)

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/7c1e92f19cd51226-1674116974338.jpg" alt="" />
Find the domain of ƒ/g.

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=2/x, g(x)=x+1

In Exercises 82–84, find f + g, f - g, fg, and f/g. f(x) = x^2 + x + 1, g(x) = x^2 -1

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ+g)(3)

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘g)(4)

Finding the Domain of a Rational Function / Fraction with Variable in Denominator

In Exercises 89–90, express the given function h as a composition of two functions f and g so that h(x) = (f ○ g)(x). h(x) = (x^2 + 2x - 1)^4

In Exercises 91–94, use the graphs of f and g to evaluate each composite function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/00309d7f879a65ae-1674118632212.jpg" alt="" /> (fog) (-1)

In Exercises 31–50, find fg and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 17

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/4ac420317e274b77-1673630859369.jpg" alt="" />
Graph f-g.

For the pair of functions defined, find (ƒg)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-7

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = 1/(2x-3)

Domain of a Composition of Functions, Example 3 - Common Mistake

For the pair of functions defined, find (ƒ-g)(x).Give the domain of each. See Example 2. ƒ(x)=√(4x-1), g(x)=1/x

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1/x

Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (f ○ g)(-6)

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=x^3, g(x)=x^2+3x-1

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (g∘g)(-2)

In Exercises 31–50, find f/g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

In Exercises 59-64, let
f(x) = 2x - 5 
g(x) = 4x - 1
h(x) = x² + x + 2.
Evaluate the indicated function without finding an equation for the function. f(g[h (1)])

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1+2x^2

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

In Exercises 1–30, find the domain of each function. f(x) = x² - 2x - 15

In Exercises 31–50, find fg and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x^3, g(x)=x^2+3x-1

In Exercises 1–30, find the domain of each function. f(x) = (2x+7)/(x^3 - 5x^2 - 4x+20)

For the pair of functions defined, find (ƒ-g)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-6

In Exercises 91–94, use the graphs of f and g to evaluate each composite function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/a0ebd6ffb679d959-1674118519610.jpg" alt="" /> (go f) (0)

In Exercises 1–30, find the domain of each function. h(x) = 4/(3/x - 1)

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√(x-1), g(x)=3x

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/9a4ed95d5dc43c8a-1674040900558.jpg" alt="" />
Find(g/f)(3)

In Exercises 31–50, find f−g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

Use the table to evaluate each expression, if possible.  (f/g) (0)

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=x+2, g(x)=x^4+x^2-4

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=√(x-1), g(x)=3x

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (g∘ƒ)(0)

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x) 
f(x) = x²+2, g(x) = x² – 2

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=x+3

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

❖ Function Notation ❖

In Exercises 31–50, find ƒ/g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ/g)(5)

Find the domain and range of a function

Ex 4:  Domain of a Composite Function

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

❖ Finding the Domain of a Function -  Made Easy! ❖

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=6x+2

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ/g)(-1)

Algebra, Domain and Range

Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=-6x+9, g(x)=5x+7

Use the table to evaluate each expression, if possible.  (ƒ+g)(1)

In Exercises 1–30, find the domain of each function. g(x) = 3/(x^2-2x-15)

In Exercises 1–30, find the domain of each function. f(x) = 1/(x^2+1) - 1/(x^2-1)

Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2
(ƒ∘g)(2)

For the pair of functions defined, find (ƒ/g)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

In Exercises 59-64, let
f(x) = 2x - 5 
g(x) = 4x - 1
h(x) = x² + x + 2.
Evaluate the indicated function without finding an equation for the function. (fog) (0)

❖ How to Find the Domain of a Function - Numerous Examples ❖

In Exercises 1–30, find the domain of each function. f(x) = 1/[3/(x - 1) - 2]

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x^2-4x+2

For the pair of functions defined, find (ƒ+g)(x).Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=2/x, g(x)=x+1

❖ Composition of Functions ❖

For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ+g)(-5)

Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2
(g∘ƒ)(2)

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x) 
f(x) = 4-x, g(x) = 2x² +x+5

In Exercises 76–81, find the domain of each function. f(x) = x/(x^2 + 4x -21)

In Exercises 31–50, find f/g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

Domain of a Composition of Functions

In Exercises 31–50, find f/g and determine the domain for each function. f(x) = √x, g(x) = x − 4

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ-g)(-1)

In Exercises 51–66, find 
a. (fog) (2)
b. (go f) (2) 
f(x) = 4-x, g(x) = 2x² +x+5

For the pair of functions defined, find (ƒ-g)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

For the pair of functions defined, find (ƒ+g)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-5

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/1a06c17a78d4a112-1674042876231.jpg" alt="" />
Find (f+g)(−3).

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = 2x, g(x) = x+7

In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = (3x − 1)^4

In Exercises 31–50, find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/e86f8fb7dd69de32-1673629205346.jpg" alt="" /> Graph f+g.

In Exercises 51–66, find 
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2). 
f(x) = 2x-3, g(x) = (x+3)/2

In Exercises 1–30, find the domain of each function. f(x) = 1/√(x - 3)

In Exercises 67-74, find
a. (fog) (x)
b. the domain of f o g. 
f(x) = √x, g(x) = x − 2

❖ Finding the Domain of a Function Algebraically (No graph!) ❖

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=x+3

In Exercises 31–50, find f/g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

In Exercises 31–50, find f/g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Use the graphs of f and g to solve Exercises 83–90. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/35175917040ee706-1674042630806.jpg" alt="" /> Find (g-f) (-2).

In Exercises 95–96, find all values of x satisfying the given conditions. f(x) = 2x − 5, g(x) = x² − 3x + 8, and (ƒ o g) (x) = 7.

In Exercises 76–81, find the domain of each function. g(x) = 4/(x - 7)

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘g)(-2)

In Exercises 31–50, find f−g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

Domain of a Composition of Functions, Example 1

In Exercises 67-74, find
a. (fog) (x)
b. the domain of f o g.
f(x) = 2/(x+3), g(x) = 1/x

Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (ƒ ○ g)(x)

In Exercises 1–30, find the domain of each function. f(x) = 1/(x+7) + 3/(x-9)

In Exercises 31–50, find ƒ-g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=2-x

In Exercises 31–50, find fg and determine the domain for each function. f(x) = √x, g(x) = x − 4

Rational Expressions and Domain

In Exercises 1–30, find the domain of each function. f(x) = √(5x+35)

Finding domain of composite functions

Expressing a function as a composition of two functions

Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (g ○ ƒ)(x)

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘ƒ)(2)

In Exercises 51–66, find 
c. (fog) (2)
d. (go f) (2).
f(x) = √x, g(x) = x − 1

Finding Functions that Form a Particular Composite Function

Functions: Adding and Subtracting

In Exercises 1–30, find the domain of each function. g(x) = √(x −2)/(x-5)

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