Problem 5 - College Algebra with Modeling and Visualization

Question

If \(f(x)\) calculates the number of square feet in \(x\) square yards and \(g(x)\) calculates the cost in dollars of \(x\) square feet of carpet, what does \((g \circ f)\) calculate?

Step-by-Step Solution

Answer

It calculates the cost in dollars of \(x\) square yards of carpet.

Step 1: Understand f(x)

The function \(f(x)\) is defined to compute the number of square feet from \(x\) square yards. We know that there are 9 square feet in one square yard. Therefore, \(f(x) = 9x\).

Step 2: Understand g(x)

The function \(g(x)\) is defined to calculate the cost in dollars of \(x\) square feet of carpet. Typically, this function will multiply the number of square feet by a constant rate \(c\) (the cost per square foot). Therefore, \(g(x) = cx\), where \(c\) is the cost per square foot.

Step 3: Formulate (g ̃∘ f)(x)

The composition \((g \circ f)(x)\) combines both functions by first applying \(f(x)\) and then \(g(x)\) to the result. Thus, \((g \circ f)(x) = g(f(x))\).

Step 4: Substitute f(x) into g(x)

We substitute \(f(x) = 9x\) into \(g(x)\). Therefore, \((g \circ f)(x) = g(9x) = c(9x) = 9cx\).

Step 5: Interpret the Result

The composition \((g \circ f)(x) = 9cx\) represents the cost in dollars of \(x\) square yards of carpet, given that the cost per square foot is \(c\). The function considers converting square yards to square feet before calculating the cost.

Key Concepts

Square Feet Conversion

When you're dealing with different measurement units, converting from one unit to another becomes crucial. In this exercise, the conversion is from square yards to square feet. Since there are exactly 9 square feet in a square yard, this conversion can be understood as multiplying the number of square yards by 9.

Think of it like this: if you have 2 square yards, converting to square feet involves simple multiplication:

• First, identify how many square feet are in one square yard, which is 9.
• Next, multiply the number of square yards by 9.
• For 2 square yards: \(f(x) = 9 \times 2 = 18\) square feet.

This function, represented as \(f(x) = 9x\), applies to any value of \(x\) that represents square yards. It helps in visualizing how much flooring area you truly have when measured in square feet.

Cost Calculation

Understanding cost calculation is pivotal when planning expenses. In terms of this exercise, the function \(g(x)\) is all about determining the cost of a certain number of square feet of carpet. Every square foot has a cost, represented as \(c\). Thus, the function \(g(x)\) is given by \(g(x) = cx\).

Here's how cost calculation works in practice:

• Identify the rate or cost per square foot of the carpet, known here as \(c\).
• Multiply this cost by the number of square feet you wish to cover.
• For example, if carpet costs 5 dollars per square foot, and you need 20 square feet: \(g(x) = 5 \times 20 = 100\) dollars.

This straightforward multiplication offers clarity on total expenses needed. It ensures that the cost remains aligned with the area you are covering with the carpet.

Function Evaluation

Function evaluation involves substituting one function into another as seen in the composition \(g \circ f\). Here, this involves plugging the output from one calculation into another formula.

In the exercise, we use \(f(x)\) to convert square yards into square feet. Once we have the number of square feet, we use \(g(x)\) to calculate the cost. The evaluation \(g(f(x))\) allows us to create a direct link from square yards to total cost:

• First, convert the square yards to square feet: \(f(x) = 9x\).
• Next, apply this result in the cost function: \((g \circ f)(x) = g(9x) = 9cx\), where \(c\) is the cost per square foot.

This combined function makes it seamless to evaluate direct costs from yard-based measurements, acknowledging both area and pricing in one calculation. By using function compositions, we streamline complex calculations into simple, actionable steps.