Inverse Function: Find F⁻¹ For F(x) = 9x - 2

Nov 13, 2025 by ADMIN 45 views

Find the Inverse of the Given One-to-One Function *f*. Give the Domain and the Range of *f* and of *f⁻¹*, and then Graph Both *f* and *f⁻¹* on the Same Set of Axes. *f(x) = 9x - 2*

Alright, let's dive into finding the inverse of the function f(x) = 9x - 2, determining the domains and ranges of both f and its inverse f⁻¹, and then graphing them together. It sounds like a lot, but we'll break it down step by step.

Finding the Inverse Function f⁻¹(x)

So, how do we find the inverse of a function? The basic idea is to swap x and y and then solve for y. Let's start with our function:

f(x) = 9x - 2

Replace f(x) with y:

y = 9x - 2

Now, swap x and y:

x = 9y - 2

Next, solve for y:

x + 2 = 9y

y = (x + 2) / 9

Therefore, the inverse function f⁻¹(x) is:

f⁻¹(x) = (x + 2) / 9

Verifying the Inverse

To ensure we've found the correct inverse, we can verify it by checking if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Let's check f(f⁻¹(x)):

f(f⁻¹(x)) = f((x + 2) / 9) = 9 * ((x + 2) / 9) - 2 = (x + 2) - 2 = x

Now, let's check f⁻¹(f(x)):

f⁻¹(f(x)) = f⁻¹(9x - 2) = ((9x - 2) + 2) / 9 = (9x) / 9 = x

Both compositions result in x, so our inverse function is correct!

Determining the Domain and Range of f and f⁻¹

Now, let's figure out the domain and range for both f(x) and f⁻¹(x).

Domain and Range of f(x) = 9x - 2

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.

For the function f(x) = 9x - 2, there are no restrictions on the values of x that we can plug in. We can input any real number, and we'll get a real number output. Therefore:

Domain of f: All real numbers, or (-∞, ∞)
Range of f: All real numbers, or (-∞, ∞)

Domain and Range of f⁻¹(x) = (x + 2) / 9

Similarly, for the inverse function f⁻¹(x) = (x + 2) / 9, there are also no restrictions on the values of x. We can input any real number, and we'll get a real number output. Therefore:

Domain of f⁻¹: All real numbers, or (-∞, ∞)
Range of f⁻¹: All real numbers, or (-∞, ∞)

Important Note: Notice that the domain of f is the range of f⁻¹, and the range of f is the domain of f⁻¹. This is a general property of inverse functions.

Graphing f(x) and f⁻¹(x)

Finally, let's graph both functions on the same set of axes. To do this, we can plot a few points for each function and then draw a line through them.

Graphing f(x) = 9x - 2

This is a linear function with a slope of 9 and a y-intercept of -2. Let's find two points:

When x = 0, f(0) = 9(0) - 2 = -2. So, we have the point (0, -2).
When x = 1, f(1) = 9(1) - 2 = 7. So, we have the point (1, 7).

Plot these points and draw a line through them.

Graphing f⁻¹(x) = (x + 2) / 9

This is also a linear function. Let's find two points:

When x = -2, f⁻¹(-2) = (-2 + 2) / 9 = 0. So, we have the point (-2, 0).
When x = 7, f⁻¹(7) = (7 + 2) / 9 = 1. So, we have the point (7, 1).

Plot these points and draw a line through them.

Visual Representation

When you graph these two functions on the same axes, you'll notice that they are reflections of each other across the line y = x. This is another key property of inverse functions. The graph of f⁻¹(x) is obtained by reflecting the graph of f(x) over the line y = x. This visual confirmation reinforces our algebraic calculations and understanding of inverse functions.

Key Concepts and Properties of Inverse Functions

What is a One-to-One Function?

Before we proceed further, it's important to understand the concept of a one-to-one function. A function is one-to-one if each y-value corresponds to only one x-value. In other words, no two different x-values produce the same y-value. Graphically, a function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph more than once. Why is this important? Well, a function must be one-to-one to have an inverse. If a function is not one-to-one, swapping x and y will not result in a valid function.

The Horizontal Line Test

The horizontal line test is a visual method used to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, the function is not one-to-one. The function f(x) = 9x - 2 is a linear function and will pass the horizontal line test, confirming that it is indeed one-to-one.

Domain and Range Relationship

As mentioned earlier, the domain of a function f(x) is the range of its inverse f⁻¹(x), and the range of f(x) is the domain of f⁻¹(x). This relationship is fundamental to understanding inverse functions. When finding the inverse, you are essentially swapping the roles of x and y, so it makes sense that the input and output values are interchanged between the original function and its inverse.

Composition of Inverse Functions

Another important property of inverse functions is that their composition results in the identity function, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This property serves as a powerful check to verify that you have correctly found the inverse of a function. If the composition does not result in x, then you have likely made an error in your calculations.

Symmetry About the Line y = x

The graphs of a function and its inverse are symmetric about the line y = x. This means that if you were to fold the coordinate plane along the line y = x, the graphs of f(x) and f⁻¹(x) would perfectly overlap. This symmetry provides a visual representation of the inverse relationship between the two functions.

Common Mistakes to Avoid

Forgetting to Swap x and y

The most common mistake when finding the inverse of a function is forgetting to swap x and y before solving for y. This is a crucial step in the process, and omitting it will lead to an incorrect inverse function.

Incorrectly Solving for y

Another common mistake is making an error while solving for y after swapping x and y. It is essential to carefully apply algebraic operations to isolate y correctly. Double-check your steps and ensure you are not making any arithmetic errors.

Not Verifying the Inverse

Failing to verify the inverse function is also a common mistake. As mentioned earlier, composing the function and its inverse should result in the identity function, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Verifying the inverse will help you catch any errors in your calculations and ensure that you have found the correct inverse function.

Confusing the Inverse with the Reciprocal

It is important to distinguish between the inverse of a function and the reciprocal of a function. The inverse function f⁻¹(x) undoes the operation of the original function f(x), while the reciprocal of a function is simply 1 divided by the function, 1/f(x). These are two entirely different concepts, and confusing them can lead to incorrect answers.

Real-World Applications of Inverse Functions

Inverse functions have numerous real-world applications in various fields, including science, engineering, and economics. Here are a few examples:

Cryptography

In cryptography, inverse functions are used to encrypt and decrypt messages. The encryption process involves applying a function to the original message to transform it into an unreadable form. The decryption process involves applying the inverse function to the encrypted message to recover the original message. Think of it like a secret code. You use a function to scramble the message, and the person receiving it uses the inverse function to unscramble it.

Computer Graphics

In computer graphics, inverse functions are used to transform objects and scenes. For example, if you want to rotate an object, you can apply a rotation function to its coordinates. To undo the rotation, you can apply the inverse rotation function. This allows developers to manipulate objects in 3D space and create realistic and interactive visuals.

Economics

In economics, inverse functions are used to model supply and demand. The demand function relates the price of a product to the quantity demanded, while the supply function relates the price of a product to the quantity supplied. The inverse of the demand function gives the price as a function of quantity demanded, and the inverse of the supply function gives the price as a function of quantity supplied. Economists use these inverse functions to analyze market equilibrium and predict how prices and quantities will change in response to shifts in supply or demand.

Conclusion

So, there you have it! We successfully found the inverse of f(x) = 9x - 2, determined the domain and range of both f and f⁻¹, and discussed how to graph them. Remember the key steps: swap x and y, solve for y, and verify your answer. Understanding inverse functions is a fundamental concept in mathematics, and mastering it will open doors to more advanced topics. Keep practicing, and you'll become a pro in no time!