Problem 79 - Algebra and Trigonometry

Question

Marcello's Pizza charges a base price of \(\$ 7\) for a large pizza, plus \(\$ 2\) for each topping. Thus, if you order a large pizza with \(x\) toppings, the price of your pizza is given by the function \(f(x)=7+2 x .\) Find \(f^{-1} .\) What does the function \(f^{-1}\) represent?

Step-by-Step Solution

Answer

The function \(f^{-1}(y) = \frac{y - 7}{2}\) gives the number of toppings for a given total price.

Step 1: Understanding the Problem

We are given the function \(f(x) = 7 + 2x\), which represents the cost of a large pizza with \(x\) toppings. Our task is to find the inverse function \(f^{-1}\), which will allow us to determine the number of toppings based on the total price.

Step 2: Set the Function Equal to y

To find the inverse of a function, start by setting \(f(x)\) equal to \(y\). So, we have the equation \(y = 7 + 2x\).

Step 3: Solve for x

Rearrange the equation \(y = 7 + 2x\) to solve for \(x\). First, subtract 7 from both sides to get \(y - 7 = 2x\). Then, divide both sides by 2 to isolate \(x\), resulting in \(x = \frac{y - 7}{2}\).

Step 4: Write the Inverse Function

Now that we have \(x = \frac{y - 7}{2}\), we can express this as the inverse function: \(f^{-1}(y) = \frac{y - 7}{2}\).

Step 5: Interpret the Inverse Function

The inverse function \(f^{-1}(y)\) represents the number of toppings \(x\) that correspond to the total price \(y\) of the pizza. In other words, it tells us how many toppings were added to reach a certain total cost.

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions you will encounter in algebra. At its essence, a linear function can be identified by its format: it can be written as \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This means that the graph of a linear function is a straight line. Linear functions are fundamental in mathematics as they provide a straightforward and manageable way to model real-world situations.
In Marcello’s Pizza example, the linear function \(f(x) = 7 + 2x\) models the cost of a pizza with \(x\) toppings. Here, the base price is \(\\(7\), and \(\\)2\) is the rate at which the cost increases per topping. This linear equation helps you calculate the total cost efficiently when you know the number of toppings.

• **Direct and constant rate of change:** In a linear function, the rate of change between variables remains constant.
• **Simple arithmetic operations:** Addition and multiplication are primarily used in linear functions.

Linear functions are not just about pizza prices. They can apply to various scenarios, such as calculating wages, distance over time, and more.

Solving Equations

Solving equations is the process of finding the unknown value in an equation. It involves manipulating the equation in several steps to isolate the variable, usually on one side of the equation. In the problem of finding an inverse function, the goal is to express the dependent variable in terms of the independent variable.
Let's look at how it works in Marcello's Pizza function. Given \(y = 7 + 2x\), to solve for \(x\), follow these steps:

• Subtract 7 from both sides to get: \(y - 7 = 2x\)
• Divide both sides by 2 to isolate \(x\): \(x = \frac{y - 7}{2}\)

This rearrangement lets you turn the initial function into its inverse form, \(f^{-1}(y)\), offering a new perspective on the relationship between the number of toppings and the total price. Solving equations is a crucial skill in finding inverse functions, enabling you to source a specific input given a known output.

Function Notation

Function notation is a way to represent functions in a systematic form, making it easier to identify and work with them. In function notation, \(f(x)\) denotes a function named \(f\) applied to the variable \(x\). This notation helps clearly communicate the relationship between inputs and outputs without ambiguity.
In our example, \(f(x) = 7 + 2x\) is the function notation for the pizza pricing problem, signifying that the cost depends on the number of toppings \(x\). When we find its inverse, we use \(f^{-1}(y)\), indicating a swap of roles, where \(y\) now stands for inputs and \(f^{-1}(y)\) for outputs. Specifically, \(f^{-1}(y) = \frac{y - 7}{2}\) helps determine how many toppings were ordered to reach a total price of \(y\).

• **Avoid confusion:** Function notation allows handling different functions with precision and clarity.
• **Understanding operations:** With function notation, applying mathematical operations or transformations on functions becomes simpler and clearer.

In conclusion, understanding function notation is vital for troubleshooting and interpreting equations, especially while working with inverse functions.