Problem 72 - Algebra 2

Question

Writing A salesperson earns a 3\(\%\) bonus on weekly sales over \(\$ 5000\) . $$\begin{array}{l}{g(x)=0.03 x} \\ {h(x)=x-5000}\end{array}$$ a. Explain what each function above represents. b. Which composition, \((h \circ g)(x)\) or \((g \circ h)(x),\) represents the weekly bonus? Explain.

Step-by-Step Solution

Answer

The function \(g(x)=0.03x\) represents the bonus calculation as 3% of the sales amount. The function \(h(x)=x-5000\) represents the amount of sales over $5000. The composition that represents the weekly bonus calculation is \(g(h(x))\) because this composition first computes the sales amount over $5000, then calculate 3% bonus on that amount.

Step 1: Understanding Functions

The given functions are \(g(x)=0.03x\) and \(h(x)=x-5000\). Here, \(g(x)\) represents the bonsu being calculated as 3\% of total sales, while \(h(x)\) represents the sales amount exceeding $5000.

Step 2: About Composition

Functions composition involves combining functions such that the output of one function becomes the input into another function.

Step 3: Find the right composition

Here, we need to identify the composition that calculates the weekly bonus accurately. Considering the process of bonus calculation, first the amount over $5000 is computed and then 3% of this amount is calculated as bonus. Therefore, the correct composition would be \(g(h(x))\).

Step 4: Function Composition Representation

In the composition \(g(h(x))\), the function h(x) will be applied first which presents the amount of sales over $5000. Then, the result of h(x) will be the input to the function \(g(x)\), which calculates the 3% bonus on that amount. So, \(g(h(x))\) accurately represents the calculation of the weekly bonus.

Key Concepts

Functions and Their Roles

Functions are mathematical expressions that relate an input to an output. In this context of calculating bonuses, we have two specific functions: \( g(x) = 0.03x \) and \( h(x) = x - 5000 \). These functions each have unique roles.

• \( g(x) = 0.03x \): This function calculates 3\(\%\) of a given value. Here, it is used to find what 3\(\%\) of the sales amount would be.
• \( h(x) = x - 5000 \): This function determines how much the sales exceed $5000. It helps in identifying if the sales overstep the threshold for bonus eligibility.

These functions are tools utilized for different, specific purposes in the bonus calculation process.

Algebra and Function Composition

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It serves as the foundation for understanding how we combine and use functions. When working with functions like \( g(x) \) and \( h(x) \), composition is key.
Function composition involves creating a new function by applying one function to the result of another. This is expressed as \((f \circ g)(x)\), which means you first apply \(g(x)\) and then \(f(x)\).

• In this exercise, we have to find which function composition accurately calculates the bonus.
• The right composition transforms the calculated sales over $5000 into the bonus amount with the 3\(\%\) rate.

Understanding algebraic functions and their compositions is crucial because it connects the operational sequence to the desired outcome.

Calculating the Bonus

The process of calculating a bonus involves multiple steps, each clearly represented by the functions and their composition.
The primary goal is to compute the bonus based on sales that exceed \(5000. Here's how it's done:

• Firstly, compute how much the sales exceed the \)5000 mark using \( h(x) = x - 5000 \). This sets the stage for determining the bonus amount.
• Next, apply the function \( g \) to this result — \( g(h(x)) = g(x - 5000) = 0.03(x - 5000) \) — to find 3\(\%\) of the amount that exceeds $5000, effectively giving us the bonus.

This composition, \( g(h(x)) \), accurately represents the process of calculating the weekly sales bonus.