Problem 89 - Algebra and Trigonometry

Question

Pizza Cost 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 inverse function is \(f^{-1}(y) = \frac{y - 7}{2}\); it represents the number of toppings given a total pizza price of \(y\).

Step 1: Understand the Inverse Function Concept

The inverse function, denoted as \(f^{-1}\), is a function that "undoes" the operation of the original function. For a given function \(f(x)\), the inverse \(f^{-1}(y)\) will take the value \(y\) (the output of \(f(x)\)) and map it back to \(x\) (the input of \(f(x)\)).

Step 2: Set the Function Expression

We are given the function \(f(x) = 7 + 2x\). Our goal is to express \(x\) in terms of \(y\), where \(y = f(x)\). Essentially, find \(x\) that satisfies the equation \(y = 7 + 2x\).

Step 3: Solve the Equation for x

Given \(y = 7 + 2x\), we will solve for \(x\):Subtract 7 from both sides to isolate terms with \(x\): \(y - 7 = 2x\).Next, divide both sides by 2 to solve for \(x\): \(x = \frac{y - 7}{2}\).

Step 4: Define the Inverse Function

Now that we have solved for \(x\) in terms of \(y\), the inverse function can be expressed as:\[f^{-1}(y) = \frac{y - 7}{2}\].This function will give the quantity of toppings \(x\) based on the total price \(y\).

Step 5: Interpret the Meaning of the Inverse

The inverse function \(f^{-1}(y)\) represents the number of toppings \(x\) on a pizza that results in a total price of \(y\). In other words, given the total cost of a pizza, \(f^{-1}\) helps determine how many toppings were ordered to reach that price.

Key Concepts

Function Notation

Function notation is a systematic way to represent functions in mathematics. It's like giving each function a clear label that tells us how to handle its inputs and outputs. In the exercise we just reviewed, a function is given in the form of \( f(x) = 7 + 2x \). Here, \( f \) is the function name, and \( x \) is the variable or letter that stands in for any number you choose.

When you see \( f(x) \), you plug in a value for \( x \), and the expression gives you back a number, which is called the output. Function notation is useful because it makes it easy to see that we're working with a function and not just a regular equation.

• It helps us understand how to calculate new outputs by substituting different input values.
• Function notation is especially beneficial in more complex problems because it conveys precise relationships between variables.
  

Understanding this notation is essential because it lays the foundation for comprehending more advanced topics like the inverse function we solved.

Mathematical Modelling

Mathematical modelling involves creating mathematical expressions or functions to simulate real-world scenarios, such as the cost of a pizza.

In the original exercise, we developed a cost function for a pizza using the equation \( f(x) = 7 + 2x \), which models the total cost based on the number of toppings \( x \).

• The base price, \( 7 \), represents the fixed cost of a large pizza without toppings.
• The \( 2x \) indicates the variable cost, changing based on the quantity of toppings.

Mathematical models like this allow us to predict, understand, and make informed decisions based on changing variables in practical situations. Once a model is established, we can use it to examine different scenarios, such as calculating the price for any number of toppings or solving it backwards to determine inputs from known results.

Solving Equations

Solving equations is a key skill in math that involves finding the unknown variable's value that makes the equation true. In the exercise, we had the initial equation \( y = 7 + 2x \), where we needed to solve for \( x \).

This process involves rearranging the equation to isolate \( x \) on one side. We started by subtracting 7 from both sides: \[ y - 7 = 2x \]Then, we divided both sides by 2 to find \( x \): \[ x = \frac{y - 7}{2} \] This rearrangement lets us solve for any number of toppings \( x \) given the total price \( y \).

• Solving equations generally involves reversing the operations done to the variable.
• Understanding this concept is vital, as it translates across all types of equations and mathematical contexts.

In this example, solving the equation was crucial for finding the inverse function \( f^{-1}(y) \), which determines how many toppings correspond to a specific price. Mastering this technique helps you tackle a wide array of mathematical challenges.