Determining Whether the Functions Are Inverses

Oct 13, 2025 by wbrum.org 93 views

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Introduction

In mathematics, functions are used to describe relationships between variables. When we have two functions, we can compose them to create a new function. This process involves plugging one function into the other, and it's an essential concept in algebra and calculus. In this article, we'll explore how to find the compositions of two given functions, $$f(x)$$ and $$g(x)$$, and determine whether they are inverses of each other.

The Given Functions

We are given two functions:

f(x) = 9x - 5

g(x) = \frac{x + 9}{5}

Finding $$f(g(x))$$

To find the composition $$f(g(x))$$, we need to plug the function $$g(x)$$ into the function $$f(x)$$. This means we'll replace $$x$$ in the function $$f(x)$$ with the expression $$g(x)$$.

f(g(x)) = 9\left(\frac{x + 9}{5}\right) - 5

To simplify this expression, we'll start by multiplying $$9$$ by the fraction:

f(g(x)) = \frac{9(x + 9)}{5} - 5

Next, we'll distribute the $$9$$ to the terms inside the parentheses:

f(g(x)) = \frac{9x + 81}{5} - 5

Now, we'll subtract $$5$$ from the fraction:

f(g(x)) = \frac{9x + 81}{5} - \frac{25}{5}

To combine the fractions, we'll find a common denominator, which is $$5$$:

f(g(x)) = \frac{9x + 81 - 25}{5}

Simplifying the numerator, we get:

f(g(x)) = \frac{9x + 56}{5}

Finding $$g(f(x))$$

To find the composition $$g(f(x))$$, we need to plug the function $$f(x)$$ into the function $$g(x)$$. This means we'll replace $$x$$ in the function $$g(x)$$ with the expression $$f(x)$$.

g(f(x)) = \frac{(9x - 5) + 9}{5}

To simplify this expression, we'll start by combining the terms inside the parentheses:

g(f(x)) = \frac{9x - 5 + 9}{5}

Next, we'll simplify the numerator:

g(f(x)) = \frac{9x + 4}{5}

Determining Whether the Functions Are Inverses

To determine whether the functions $$f(x)$$ and $$g(x)$$ are inverses of each other, we need to check if the compositions $$f(g(x))$$ and $$g(f(x))$$ are equal to the identity function, which is $$x$$.

We've already found the compositions:

f(g(x)) = \frac{9x + 56}{5}

g(f(x)) = \frac{9x + 4}{5}

To check if these compositions are equal to the identity function, we'll set them equal to $$x$$ and solve for $$x$$.

First, let's set $$f(g(x))$$ equal to $$x$$:

\frac{9x + 56}{5} = x

To solve for $$x$$, we'll multiply both sides by $$5$$:

9x + 56 = 5x

Next, we'll subtract $$5x$$ from both sides:

4x + 56 = 0

Now, we'll subtract $$56$$ from both sides:

4x = -56

Finally, we'll divide both sides by $$4$$:

x = -14

This means that $$f(g(x))$$ is not equal to the identity function $$x$$.

Next, let's set $$g(f(x))$$ equal to $$x$$:

\frac{9x + 4}{5} = x

To solve for $$x$$, we'll multiply both sides by $$5$$:

9x + 4 = 5x

Next, we'll subtract $$5x$$ from both sides:

4x + 4 = 0

Now, we'll subtract $$4$$ from both sides:

4x = -4

Finally, we'll divide both sides by $$4$$:

x = -1

This means that $$g(f(x))$$ is not equal to the identity function $$x$$.

Since neither composition is equal to the identity function, we can conclude that the functions $$f(x)$$ and $$g(x)$$ are not inverses of each other.

Conclusion

In this article, we found the compositions $$f(g(x))$$ and $$g(f(x))$$ of the given functions $$f(x) = 9x - 5$$ and $$g(x) = \frac{x + 9}{5}$$. We then determined that neither composition is equal to the identity function, which means that the functions $$f(x)$$ and $$g(x)$$ are not inverses of each other. This demonstrates the importance of carefully evaluating the compositions of functions and understanding the properties of inverse functions.

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Introduction

In our previous article, we explored how to find the compositions of two given functions, $$f(x)$$ and $$g(x)$$, and determine whether they are inverses of each other. We found that the compositions $$f(g(x))$$ and $$g(f(x))$$ are not equal to the identity function, which means that the functions $$f(x)$$ and $$g(x)$$ are not inverses of each other.

In this article, we'll answer some common questions related to finding compositions of functions and determining inverses.

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

A: A function and its inverse are two functions that "undo" each other. In other words, if we apply a function to a value and then apply its inverse to the result, we should get back to the original value. For example, if we have a function $$f(x) = 2x + 3$$, its inverse is $$f^{-1}(x) = \frac{x - 3}{2}$$. If we apply $$f(x)$$ to a value and then apply $$f^{-1}(x)$$ to the result, we should get back to the original value.

Q: How do I know if two functions are inverses of each other?

A: To determine if two functions are inverses of each other, you need to check if the compositions of the functions are equal to the identity function. In other words, you need to check if $$f(g(x)) = x$$ and $$g(f(x)) = x$$. If both compositions are equal to the identity function, then the functions are inverses of each other.

Q: What is the identity function?

A: The identity function is a function that maps every value to itself. In other words, if we apply the identity function to a value, we get back the same value. The identity function is often denoted as $$i(x) = x$$.

Q: How do I find the composition of two functions?

A: To find the composition of two functions, you need to plug one function into the other. In other words, you need to replace the variable in one function with the expression of the other function. For example, if we have two functions $$f(x) = 2x + 3$$ and $$g(x) = x^2 + 1$$, we can find the composition $$f(g(x))$$ by plugging $$g(x)$$ into $$f(x)$$.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs (called the domain) and a set of outputs (called the range). A relation is a set of ordered pairs that satisfy a certain condition. In other words, a function is a special type of relation where each input corresponds to exactly one output.

Q: How do I determine if a function is one-to-one or onto?

A: To determine if a function is one-to-one or onto, you need to check if each input corresponds to exactly one output and if each output is reached by at least one input. If a function is one-to-one, it means that each input corresponds to exactly one output. If a function is onto, it means that each output is reached by at least one input.

Q: What is the significance of finding the composition of functions?

A: Finding the composition of functions is an important concept in mathematics because it allows us to analyze the behavior of functions and determine if they are inverses of each other. It also helps us to understand the properties of functions, such as one-to-one and onto.

Q: How do I apply the concept of composition of functions in real-life scenarios?

A: The concept of composition of functions is widely used in many real-life scenarios, such as:

In physics, the composition of functions is used to describe the motion of objects.
In engineering, the composition of functions is used to design and analyze complex systems.
In economics, the composition of functions is used to model the behavior of economic systems.

Conclusion

In this article, we answered some common questions related to finding compositions of functions and determining inverses. We hope that this article has provided you with a better understanding of the concept of composition of functions and its significance in mathematics and real-life scenarios.