Problem 98 - College Algebra with Modeling and Visualization

Question

Converting Units There are 4 quarts in 1 gallon, 4 cups in 1 quart, and 16 tablespoons in 1 cup. (a) Write a function \(Q\) that converts \(x\) gallons to quarts. (b) Write a function \(C\) that converts \(x\) quarts to cups. (c) Write a function \(\mathbf{T}\) that converts \(\mathbf{x}\) cups to tablespoons. (d) Express a function \(F\) that converts \(x\) gallons to tablespoons as a composition of three functions. (e) Write a formula for \(\mathbf{F}\).

Step-by-Step Solution

Answer

(a) \( Q(x) = 4x \); (b) \( C(x) = 4x \); (c) \( T(x) = 16x \); (d) \( F(x) = 256x \).

Step 1: Convert Gallons to Quarts

Given that there are 4 quarts in 1 gallon, the function \( Q(x) \) that converts \( x \) gallons to quarts is defined as:\[ Q(x) = 4x \]

Step 2: Convert Quarts to Cups

Since 1 quart is equivalent to 4 cups, the function \( C(x) \) for converting \( x \) quarts to cups is:\[ C(x) = 4x \]

Step 3: Convert Cups to Tablespoons

With 16 tablespoons in 1 cup, the function \( T(x) \) that converts \( x \) cups to tablespoons is:\[ T(x) = 16x \]

Step 4: Compose Functions for Gallons to Tablespoons

To find the function \( F(x) \) that converts gallons to tablespoons, combine the results from previous steps into a composition:\[ F(x) = T(C(Q(x))) \]

Step 5: Write Formula for Composed Function

Substitute the expressions from Steps 1, 2, and 3 into the composition to get the formula for \( F(x) \):\[ F(x) = T(C(Q(x))) = T(C(4x)) = T(4(4x)) = T(16x) = 16(16x) = 256x \]

Key Concepts

Function Composition

Function composition is a mathematical operation where you combine different functions to create a new function. It's like stacking operations in a specific order. Imagine solving a puzzle by putting pieces together. In mathematics, you use the output of one function as the input of another, creating a chain of calculations.

Let's break this down using our exercise. We had three separate functions:

• Function \( Q(x) = 4x \) converts gallons to quarts.
• Function \( C(x) = 4x \) converts quarts to cups.
• Function \( T(x) = 16x \) converts cups to tablespoons.

By composing these functions, we found a way to convert gallons directly to tablespoons: \( F(x) = T(C(Q(x))) \).

This means we first convert gallons to quarts using \( Q(x) \), then quarts to cups using \( C(x) \), and finally cups to tablespoons using \( T(x) \). Function composition allows us to see the entire conversion process as a single operation.

Quadratic Functions

Although not explicitly covered in the exercise, understanding functions is easier when we know about specific types like quadratic functions. Quadratic functions are a special class of polynomials characterized by the term \( ax^2 \). They often appear as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

Quadratic functions have a distinctive U-shaped graph known as a parabola. These functions are used in various applications like physics to model projectile motion. While our solution focused on linear functions, where changes are proportional to \( x \), recognizing quadratic forms helps in differentiating more complex function types.

In practice, quadratic functions involve cases where outputs change in relation to the square of the input, unlike the simple multipliers in unit conversion where everything scales linearly.

Mathematical Functions

Mathematical functions are foundational in expressing relationships between numbers. They represent a rule that assigns each input exactly one output. In our unit conversion exercise, each function effectively captures a step in the conversion process.

To construct a function, consider:

• The domain (set of input values).
• The rule (how to calculate the output from the input).
• The range (possible output values).

Each function we used, like \( Q(x) \), \( C(x) \), and \( T(x) \), follows this structure.

This consistent input-output relationship illustrates not just unit conversion but also provides a broader mathematical understanding, showcasing how functions map inputs to outputs in countless scenarios.