Jake Earns $ 7.50 \$7.50 $7.50 Per Hour Working At A Local Car Wash. The Function F ( X ) = 7.50 X F(x) = 7.50x F ( X ) = 7.50 X Relates The Amount Jake Earns To The Number Of Hours He Works. Write The Inverse Of This Relation.

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The Inverse of a Linear Function: Unlocking Jake's Earnings

In the world of mathematics, functions are used to describe the relationship between variables. A function can be thought of as a machine that takes an input, performs some operation on it, and produces an output. In this article, we will explore the concept of inverse functions, specifically the inverse of a linear function. We will use the example of Jake, who earns $7.50\$7.50 per hour working at a local car wash, to illustrate the concept.

The function f(x)=7.50xf(x) = 7.50x relates the amount Jake earns to the number of hours he works. This function is a linear function, which means it has a constant rate of change. In this case, the rate of change is $7.50\$7.50 per hour. The function can be graphed as a straight line, with the x-axis representing the number of hours worked and the y-axis representing the amount earned.

The inverse of a function is a function that undoes the action of the original function. In other words, if we apply the inverse function to the output of the original function, we get back the original input. To find the inverse of the function f(x)=7.50xf(x) = 7.50x, we need to swap the x and y variables and solve for y.

Let's start by writing the function as y=7.50xy = 7.50x. To find the inverse, we swap the x and y variables, which gives us x=7.50yx = 7.50y. Now, we need to solve for y.

To solve for y, we can divide both sides of the equation by 7.50, which gives us:

x7.50=y\frac{x}{7.50} = y

This can be rewritten as:

y=x7.50y = \frac{x}{7.50}

The inverse function is f−1(x)=x7.50f^{-1}(x) = \frac{x}{7.50}. This function takes the amount earned as input and produces the number of hours worked as output.

The graph of the inverse function is a reflection of the graph of the original function across the line y = x. This means that if we have a point (a, b) on the graph of the original function, the corresponding point on the graph of the inverse function is (b, a).

Let's say Jake works for 5 hours and earns $37.50\$37.50. We can use the original function to find the amount earned, and then use the inverse function to find the number of hours worked.

Using the original function, we get:

f(5)=7.50(5)=37.50f(5) = 7.50(5) = 37.50

Using the inverse function, we get:

f−1(37.50)=37.507.50=5f^{-1}(37.50) = \frac{37.50}{7.50} = 5

This makes sense, since we know that Jake worked for 5 hours.

In this article, we explored the concept of inverse functions, specifically the inverse of a linear function. We used the example of Jake, who earns $7.50\$7.50 per hour working at a local car wash, to illustrate the concept. We found the inverse function by swapping the x and y variables and solving for y. We also graphed the inverse function and provided an example of how to use it to find the number of hours worked given the amount earned.

Inverse functions are an important concept in mathematics, as they allow us to solve equations and model real-world situations. In the example above, we used the inverse function to find the number of hours worked given the amount earned. This is just one example of how inverse functions can be used to solve problems.

Inverse functions have many real-world applications, including:

  • Physics: Inverse functions are used to model the motion of objects, such as the trajectory of a projectile.
  • Engineering: Inverse functions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Inverse functions are used to model the behavior of economic systems, such as the supply and demand of goods and services.

Q: What is an inverse function?

A: An inverse function is a function that undoes the action of the original function. In other words, if we apply the inverse function to the output of the original function, we get back the original input.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to swap the x and y variables and solve for y. This can be done by using algebraic manipulations, such as dividing both sides of the equation by a constant.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two different functions that are related to each other. The original function takes an input and produces an output, while the inverse function takes the output of the original function and produces the original input.

Q: Can a function have more than one inverse?

A: No, a function can only have one inverse. If a function has more than one inverse, it is not a one-to-one function, and the concept of an inverse function does not apply.

Q: How do I graph the inverse of a function?

A: To graph the inverse of a function, you need to reflect the graph of the original function across the line y = x. This means that if you have a point (a, b) on the graph of the original function, the corresponding point on the graph of the inverse function is (b, a).

Q: What are some real-world applications of inverse functions?

A: Inverse functions have many real-world applications, including:

  • Physics: Inverse functions are used to model the motion of objects, such as the trajectory of a projectile.
  • Engineering: Inverse functions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Inverse functions are used to model the behavior of economic systems, such as the supply and demand of goods and services.

Q: Can I use inverse functions to solve equations?

A: Yes, inverse functions can be used to solve equations. By applying the inverse function to both sides of the equation, you can isolate the variable and solve for it.

Q: How do I use inverse functions to model real-world situations?

A: To use inverse functions to model real-world situations, you need to identify the input and output variables, and then use the inverse function to relate them. For example, if you are modeling the motion of an object, you can use the inverse function to relate the position and velocity of the object.

Q: What are some common mistakes to avoid when working with inverse functions?

A: Some common mistakes to avoid when working with inverse functions include:

  • Not checking if the function is one-to-one: If the function is not one-to-one, the concept of an inverse function does not apply.
  • Not swapping the x and y variables: Failing to swap the x and y variables can lead to incorrect results.
  • Not solving for y: Failing to solve for y can lead to incorrect results.

In conclusion, inverse functions are an important concept in mathematics that have many real-world applications. By understanding how to find and graph the inverse of a function, you can use it to solve equations and model real-world situations. Remember to check if the function is one-to-one, swap the x and y variables, and solve for y to avoid common mistakes.