AP Precalculus Section 2.7 Example: Domain of a Composite Function

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

Finding the domain of a composite function involving rational equations involves determining the restrictions on the variables that would make the function undefined. Here's a brief process:

1. **Identify the composite function:** Given two functions, let's say \( f(x) \) and \( g(x) \), the composite function is usually represented as \( (f \circ g)(x) \) or \( f(g(x)) \).

2. **Find the individual function domains:** Start by finding the domain of each function involved in the composite function. For rational equations, look out for denominators. Any value that would make a denominator zero should be excluded from the domain.

3. **Combine the domains:** After finding the individual domains of \( f(x) \) and \( g(x) \), consider the restrictions on the variable \( x \) that apply to both functions. The domain of the composite function is the overlap or intersection of these individual domains.

4. **Simplify if necessary:** If the composite function involves simplification steps, ensure that the restrictions identified in step 3 still apply after simplification. Adjust the domain accordingly.

Remember, for rational equations, excluded values generally arise from denominators being equal to zero. So, when finding the domain of a composite function involving rational equations, pay special attention to those denominators in each function and their restrictions on the variable \( x \).

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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