AP Precalculus Session 5

Calculator Setup: The task model requires a graphing calculator. It is recommended to set the calculator to radian mode. While Unit 1 and Unit 2 tasks do not strictly require it, Unit 3 (trigonometry) will, and having it set early ensures consistency.
Contextual Presentation: This specific type of question is not presented within a real-world context.
Function Representations: Functions are presented in three ways: * Graphically: Using a visual representation of the function. * Numerically: Using a tabular format (tables). * Analytically: Using a defined mathematical formula or equation.
AP Exam Structure: Students typically receive one analytically defined function and either a graphical or numerical (tabular) function. This model includes three parts requiring concepts from Unit 1 and Unit 2.

Problem Statement: Given $h(x) = g(f(x))$ , find the value of $h(10)$ . Provide a decimal approximation or indicate if it is not defined.
The Assembly Line Analogy: Composite functions function like an assembly line where the input goes through a sequence of steps: 1. Start with the original input ( $x = 10$ ). 2. Plug the input into the inner function ( $f$ ). 3. Take the output of the inner function and use it as the input for the outer function ( $g$ ). 4. Obtain the final output.
Step-by-Step Calculation for $h(10)$ : * Step 1: Find $f(10)$ . Referring to the provided table, the output for an input of $10$ in function $f$ is $-2$ . * Step 2: Use the output from Step 1 as the input for $g$ . Substitute $-2$ into the analytical definition: $g(x) = 5.51 \times 1.03^x$ . * Step 3: Calculation: $g(-2) = 5.51 \times 1.03^{-2}$ .
Final Required Accuracy: On the AP exam, final answers must be accurate to the thousandths place. * Truncation: Taking the first three digits after the decimal point without rounding (e.g., $5.113$ if applicable). * Rounding: Looking at the fourth decimal place to determine if the third should increase or stay the same. * Result: The calculated value for $h(10)$ is $5.114$ (accepted value $5.113$ or $5.114$ ).

Case 1: Numerical Function ( $f(x) = 5$ ): * Task: Identify all values of $x$ for which the output of $f(x)$ is $5$ . * Procedure: Review the table for the row containing output values of $f(x)$ . Look for the value $5$ . * Observation: The output $5$ appears twice in the table. * Solution: The input values corresponding to an output of $5$ are $x = 2$ and $x = 12$ .
Case 2: Analytical Function ( $g(x) = -20$ ): * Task: Find all values of $x$ as decimal approximations for which $g(x) = -20$ . * Analytical Reasoning: The function is defined as $g(x) = 5.51 \times 1.03^x$ . Because the leading coefficient is positive ( $5.51$ ) and the base is positive ( $1.03$ ), the product of positive factors will always be positive. Therefore, the function can never produce a negative output. * Graphical Reasoning: If one were to graph $y = g(x)$ and the horizontal line $y = -20$ , the lines would never intersect because the output values of $g$ are always positive. * Solution: There are no such values for which $g(x) = -20$ .

Terminology: The phrase "decreases without bound" is the mathematical equivalent of stating that $x$ approaches negative infinity ( $x \rightarrow -\infty$ ).
Notation: End behavior is formally expressed using limit notation.
Task: Determine the end behavior of $g$ as $x$ decreases without bound.
Limit Expression: $\lim_{x \to -\infty} g(x)$ .
Evaluation: Substituting the function, we find $\lim_{x \to -\infty} 5.51 \times 1.03^x$ .
Result: As $x$ approaches negative infinity for this exponential growth function, the output approaches $0$ . This can be confirmed visually via the graph of the function.

Definition of Invertibility: A function $f$ is invertible (has an inverse function) on a specified domain if each output value is mapped from a unique input value. This is known as being "one-to-one."
Criteria for Failure: If a function has repeated output values for different input values, it is not invertible.
Analysis of Function $f$ : * Reviewing the table for $f$ shows repeated output values. * Example 1: $f(2) = 5$ and $f(12) = 5$ . The output value $5$ is not mapped to a unique input. * Example 2: $f(4) = -2$ and $f(10) = -2$ . The output value $-2$ is not mapped to a unique input.
Conclusion: Since there are outputs that are not mapped to unique input values, $f$ is not an invertible function.

Evaluation of Outputs for Given Inputs: For numerical functions, identifying which inputs provide a specific output requires reviewing the function's table. In contrast to analytical functions, numerical outputs can yield multiple corresponding inputs, illustrating the unique nature of each function's representation.
Use of Contextual Presentation: Certain mathematical questions may not be framed in a real-world context, which highlights the need for understanding the abstract mathematical principles at play rather than relying on applied scenarios. This conceptual clarity is crucial for problem-solving in purely theoretical situations.

Calculator Setup: A graphing calculator set to radian mode is essential, particularly for Unit 3 (trigonometry) tasks, ensuring consistent computation.
Function Representations: Functions can be represented graphically, numerically (tables), or analytically (mathematical formulas), illustrating different ways to understand mathematical concepts.
Composite Functions: The evaluation of composite functions follows a sequential process, similar to an assembly line, highlighting the importance of thorough step-by-step calculations.
Invertibility: A function is invertible only if each output value corresponds to a unique input value, emphasizing the concept of one-to-one functions in function analysis.
Limit Notation: Understanding end behavior is vital in limits, demonstrated by the evaluation of functions as variables approach infinity, linking calculus concepts to practical scenarios.