3. Functions and Graphs

Combine Functions Using the Algebra of Function

Combinations of Functions; Composite Functions

Functions: Adding and Subtracting

Express A Function As A Composition Of Two Functions

Learn How to Compose Two Rational Functions and Simplify

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

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

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

Combining functions by adding and subtracting

Understand restrictions on functions in terms of domain and range

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

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 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

Domain of a Composition of Functions, Example 2

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)

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

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

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

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

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

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

❖ Function Notation ❖

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! ❖

Algebra, Domain and Range

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)

❖ 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]

❖ Composition of Functions ❖

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)

Domain of a 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 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 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!) ❖

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)

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

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

Rational Expressions and Domain

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)

Finding domain of composite functions

Expressing a function as a composition of two functions

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

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College Algebra

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1. Fundamental Concepts of Algebra

2. Equations and Inequalities

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions