Step-by-Step Guidance for Combining Functions (Calculus Study Prep)

Q1. Which of the following pairs of $u(x)$ and $v(x)$ is correct for $u(v(x)) = (x^3 - 5)^4$?

Background

Topic: Function Composition

This question tests your understanding of how to decompose a composite function into its inner and outer functions. You need to identify which functions $u(x)$ and $v(x)$, when composed as $u(v(x))$, produce the given expression.

Key Terms and Formulas:

• Function composition: $u(v(x))$ means you substitute $v(x)$ into $u(x)$ wherever $x$ appears.

• Outer function: $u(x)$

• Inner function: $v(x)$

Step-by-Step Guidance

1. Examine the composite function $u(v(x)) = (x^3 - 5)^4$ and identify the "outer" and "inner" parts.

2. Notice that the expression is a power function applied to a polynomial: the "outer" function is something raised to the fourth power, and the "inner" function is $x^3 - 5$.

3. Write $u(x)$ as the function that takes an input and raises it to the fourth power: $u(x) = x^4$ or $u(x) = (x)^4$.

4. Write $v(x)$ as the function that produces $x^3 - 5$.

5. Check which answer choices match this setup, but stop before selecting the final answer.

Try solving on your own before revealing the answer!

Final Answer: u(x) = x^4, v(x) = x^3 - 5

When you substitute $v(x) = x^3 - 5$ into $u(x) = x^4$, you get $u(v(x)) = (x^3 - 5)^4$.

Q2. The graphs of $u(x)$ and $v(x)$ are shown. Find the value of $u(v(3))$.

Background

Topic: Function Composition and Graph Interpretation

This question tests your ability to evaluate a composite function using information from graphs. You need to find $v(3)$ from the graph of $v(x)$, then use that value to find $u(v(3))$ from the graph of $u(x)$.

Key Terms and Formulas:

• Composite function: $u(v(x))$

• Graph interpretation: Reading values from a graph

Step-by-Step Guidance

1. Locate $x = 3$ on the graph of $v(x)$ and determine the value $v(3)$.

2. Take the value you found for $v(3)$ and use it as the input for $u(x)$.

3. Locate this input value on the graph of $u(x)$ and determine $u(v(3))$.

4. Compare your result to the answer choices, but stop before stating the final value.

Try solving on your own before revealing the answer!

Final Answer: 1

By reading the graphs, $v(3)$ gives a certain value, and $u$ evaluated at that value gives 1.

Q3. The following table is given for some values of $u(x)$ and $v(x)$, where $u(x)$ is an even function and $v(x)$ is an odd function. Find the value of $v(v(-3))$.

Background

Topic: Function Composition, Even and Odd Functions

This question tests your understanding of function composition and properties of even and odd functions. You need to use the table and the definitions of even and odd functions to evaluate $v(v(-3))$.

Key Terms and Formulas:

• Even function: $u(-x) = u(x)$ for all $x$

• Odd function: $v(-x) = -v(x)$ for all $x$

• Function composition: $v(v(-3))$ means you first find $v(-3)$, then use that as the input for $v(x)$.

Step-by-Step Guidance

1. Use the property of odd functions to find $v(-3)$ using the table: $v(-3) = -v(3)$.

2. Look up $v(3)$ in the table and calculate $v(-3)$.

3. Now, use the value of $v(-3)$ as the input for $v(x)$: find $v(v(-3))$.

4. Check the table to see if the value you need is listed, or use the properties of odd functions to deduce it.

5. Compare your result to the answer choices, but stop before stating the final value.

Try solving on your own before revealing the answer!

Final Answer: -9

Using the table and the property of odd functions, $v(v(-3)) = -9$.

Q4. Consider the function $g(x) = bx(2-x)$, where $b$ is a real number and $1 \leq b \leq 3$. If we define a new function $p(x) = g(g(x))$, what is the degree of $p(x)$?

Background

Topic: Function Composition and Degree of Polynomials

This question tests your understanding of how composing polynomial functions affects their degree. You need to determine the degree of $p(x)$ when $g(x)$ is composed with itself.

Key Terms and Formulas:

• Degree of a polynomial: The highest power of $x$ in the expression.

• Function composition: $p(x) = g(g(x))$

Step-by-Step Guidance

1. Identify the degree of $g(x) = bx(2-x)$. Expand $g(x)$ to see its degree.

2. Write $g(x)$ in expanded form: $g(x) = bx \cdot 2 - bx \cdot x = 2bx - bx^2$.

3. Notice that $g(x)$ is a degree 2 polynomial.

4. Now, consider $p(x) = g(g(x))$. Substitute $g(x)$ into itself and determine the degree of the resulting expression.

5. Think about how the degree changes when you compose two degree 2 polynomials, but stop before stating the final degree.

Try solving on your own before revealing the answer!

Final Answer: 4

Composing two degree 2 polynomials results in a degree 4 polynomial.