Find $f(g(x))$ And $g(f(x))$ And Determine Whether The Pair Of Functions \$f$[/tex\] And $g$ Are Inverses Of Each Other.Given:$f(x) = 5x + 4$g(x) = \frac{x-4}{5}$a. $f(g(x)) =

Mar 1, 2025 by ADMIN 184 views

Find $f(g(x))$ and $g(f(x))$ and Determine Whether the Pair of Functions $f$ and $g$ Are Inverses of Each Other

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Two functions, $f$ and $g$, are said to be inverses of each other if they satisfy the condition $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in their respective domains. In this article, we will find $f(g(x))$ and $g(f(x))$ for the given functions $f(x) = 5x + 4$ and $g(x) = \frac{x-4}{5}$ and determine whether the pair of functions $f$ and $g$ are inverses of each other.

Finding $f(g(x))$

To find $f(g(x))$, we need to substitute $g(x)$ into the function $f(x)$. This means we will replace $x$ in the function $f(x) = 5x + 4$ with $g(x) = \frac{x-4}{5}$.

f(g(x)) = 5\left(\frac{x-4}{5}\right) + 4

Simplifying the expression, we get:

f(g(x)) = x - 4 + 4

f(g(x)) = x

Finding $g(f(x))$

To find $g(f(x))$, we need to substitute $f(x)$ into the function $g(x)$. This means we will replace $x$ in the function $g(x) = \frac{x-4}{5}$ with $f(x) = 5x + 4$.

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

Simplifying the expression, we get:

g(f(x)) = \frac{5x}{5}

g(f(x)) = x

Determining Whether the Pair of Functions $f$ and $g$ Are Inverses of Each Other

From the previous sections, we have found that $f(g(x)) = x$ and $g(f(x)) = x$. This satisfies the condition for two functions to be inverses of each other. Therefore, we can conclude that the pair of functions $f$ and $g$ are inverses of each other.

Conclusion

In this article, we have found $f(g(x))$ and $g(f(x))$ for the given functions $f(x) = 5x + 4$ and $g(x) = \frac{x-4}{5}$. We have also determined that the pair of functions $f$ and $g$ are inverses of each other. This is a crucial concept in mathematics, and understanding inverse functions is essential in solving problems in algebra, calculus, and other branches of mathematics.

Examples and Applications

Inverse functions have numerous applications in mathematics and other fields. Here are a few examples:

Inverse functions are used to solve equations and systems of equations.
Inverse functions are used to find the domain and range of a function.
Inverse functions are used to determine the symmetry of a function.
Inverse functions are used in calculus to find the derivative and integral of a function.

Tips and Tricks

Here are a few tips and tricks to help you understand and work with inverse functions:

To find the inverse of a function, you need to swap the x and y variables and then solve for y.
To check if two functions are inverses of each other, you need to verify that $f(g(x)) = x$ and $g(f(x)) = x$.
Inverse functions are used to solve equations and systems of equations.

Introduction

Inverse functions are a fundamental concept in mathematics, and understanding them is essential in solving problems in algebra, calculus, and other branches of mathematics. In this article, we will answer some frequently asked questions about inverse functions.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, then its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return 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 then solve for y. This means that if you have a function $f(x) = y$, then its inverse function $f^{-1}(x)$ will be $f^{-1}(y) = x$.

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 function $f(x)$ takes an input x and returns an output y, while its inverse function $f^{-1}(x)$ takes the output y and returns the original input x.

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

A: To check if two functions are inverses of each other, you need to verify that $f(g(x)) = x$ and $g(f(x)) = x$. If these conditions are satisfied, then the two functions are inverses of each other.

Q: What are some common applications of inverse functions?

A: Inverse functions have numerous applications in mathematics and other fields. Some common applications include:

Solving equations and systems of equations
Finding the domain and range of a function
Determining the symmetry of a function
Finding the derivative and integral of a function

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse of a function is unique, and it is denoted by $f^{-1}(x)$.

Q: Can a function be its own inverse?

A: Yes, a function can be its own inverse. This occurs when the function is a bijection, meaning that it is both one-to-one and onto.

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 will give you the graph of the inverse function.

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:

Confusing the function and its inverse
Not verifying that the two functions are inverses of each other
Not checking for domain and range restrictions