1. Precalculus Essentials
Fifth Edition
Chapter 1
Functions and Graphs
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2. 1.7 Combinations of Functions; Composite Functions

3. Objectives Find the domain of a function.
Combine functions using the algebra of functions, specifying domains.
Form composite functions.
Determine domains for composite functions.
Write functions as compositions.

4. Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in an even root, such as a square root, of a negative number.

5. Example: Finding the Domain of a Function
Solution: Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero.

6. The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions

7. Example: Combining Functions
b. The domain of (f + g)(x) The domain of f(x) has no restrictions. The domain of g(x) has no restrictions.

8. The Composition of Functions
​x is in the domain of g and
​g(x) is in the domain of f.

9. Example: Forming Composite Functions

10. Excluding Values from the Domain of (f circle g)(x) = f(g(x))
The following values must be excluded from the input x:

11. Example: Forming a Composite Function and Finding Its Domain (1 of 2)

12. Example: Forming a Composite Function and Finding Its Domain (2 of 2)

13. Example: Writing a Function as a Composition
Express h(x) as a composition of two functions: