Given The Functions:$\[ F(x)=\frac{3x-1}{4} \quad \text{and} \quad G(x)=\frac{4x+1}{3} \\]Answer The Following:51. Find \[$(F \circ G)(x)\$\].52. Find \[$(G \circ F)(x)\$\].53. Are \[$F(x)\$\] And \[$G(x)\$\]

Introduction

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. This concept is crucial in various fields, including calculus, algebra, and analysis. In this article, we will explore the composition of functions, focusing on the given functions $F(x)=\frac{3x-1}{4}$ and $G(x)=\frac{4x+1}{3}$. We will answer three questions related to the composition of these functions.

What is Composition of Functions?

The composition of functions is a process of combining two or more functions to create a new function. Given two functions $f(x)$ and $g(x)$, the composition of $f$ and $g$ is denoted by $(f \circ g)(x)$ and is defined as:

$$(f \circ g)(x) = f(g(x))$$

In other words, we first apply the function $g$ to the input $x$, and then apply the function $f$ to the result.

Finding $(F \circ G)(x)$

To find $(F \circ G)(x)$, we need to substitute $G(x)$ into $F(x)$.

$$(F \circ G)(x) = F(G(x))$$

Substituting $G(x) = \frac{4x+1}{3}$ into $F(x) = \frac{3x-1}{4}$, we get:

$$(F \circ G)(x) = F\left(\frac{4x+1}{3}\right)$$

Now, we substitute $\frac{4x+1}{3}$ into $F(x)$:

$$(F \circ G)(x) = \frac{3\left(\frac{4x+1}{3}\right)-1}{4}$$

Simplifying the expression, we get:

$$(F \circ G)(x) = \frac{4x+1-1}{4}$$

$$(F \circ G)(x) = \frac{4x}{4}$$

$$(F \circ G)(x) = x$$

Therefore, $(F \circ G)(x) = x$.

Finding $(G \circ F)(x)$

To find $(G \circ F)(x)$, we need to substitute $F(x)$ into $G(x)$.

$$(G \circ F)(x) = G(F(x))$$

Substituting $F(x) = \frac{3x-1}{4}$ into $G(x) = \frac{4x+1}{3}$, we get:

$$(G \circ F)(x) = G\left(\frac{3x-1}{4}\right)$$

Now, we substitute $\frac{3x-1}{4}$ into $G(x)$:

$$(G \circ F)(x) = \frac{4\left(\frac{3x-1}{4}\right)+1}{3}$$

Simplifying the expression, we get:

$$(G \circ F)(x) = \frac{3x-1+1}{3}$$

$$(G \circ F)(x) = \frac{3x}{3}$$

$$(G \circ F)(x) = x$$

Therefore, $(G \circ F)(x) = x$.

Are $F(x)$ and $G(x)$ Inverse Functions?

Two functions $f(x)$ and $g(x)$ are said to be inverse functions if their composition is equal to the identity function, i.e., $(f \circ g)(x) = (g \circ f)(x) = x$.

In this case, we have:

$$(F \circ G)(x) = x$$

$$(G \circ F)(x) = x$$

Since both compositions are equal to the identity function, we can conclude that $F(x)$ and $G(x)$ are inverse functions.

Conclusion

In this article, we explored the composition of functions, focusing on the given functions $F(x)=\frac{3x-1}{4}$ and $G(x)=\frac{4x+1}{3}$. We found that $(F \circ G)(x) = x$ and $(G \circ F)(x) = x$, which means that $F(x)$ and $G(x)$ are inverse functions. This result highlights the importance of composition of functions in mathematics and its applications in various fields.

Final Answer

To summarize, the final answers are:

• $(F \circ G)(x) = x$
• $(G \circ F)(x) = x$
• $F(x)$ and $G(x)$ are inverse functions.

References

• [1] "Composition of Functions" by Khan Academy
• [2] "Inverse Functions" by Math Open Reference
• [3] "Composition of Functions" by Wolfram MathWorld
  Composition of Functions: A Comprehensive Guide =====================================================

Q&A: Composition of Functions

Q1: What is the composition of functions?

A1: The composition of functions is a process of combining two or more functions to create a new function. Given two functions $f(x)$ and $g(x)$, the composition of $f$ and $g$ is denoted by $(f \circ g)(x)$ and is defined as:

$$(f \circ g)(x) = f(g(x))$$

Q2: How do I find the composition of two functions?

A2: To find the composition of two functions, you need to substitute one function into the other. For example, to find $(F \circ G)(x)$, you need to substitute $G(x)$ into $F(x)$.

Q3: What is the difference between $(F \circ G)(x)$ and $(G \circ F)(x)$?

A3: The difference between $(F \circ G)(x)$ and $(G \circ F)(x)$ is the order in which the functions are composed. $(F \circ G)(x)$ means that $G(x)$ is composed first, and then $F(x)$ is composed. On the other hand, $(G \circ F)(x)$ means that $F(x)$ is composed first, and then $G(x)$ is composed.

Q4: How do I know if two functions are inverse functions?

A4: Two functions $f(x)$ and $g(x)$ are said to be inverse functions if their composition is equal to the identity function, i.e., $(f \circ g)(x) = (g \circ f)(x) = x$.

Q5: What is the significance of composition of functions?

A5: The composition of functions is a fundamental concept in mathematics that allows us to combine two or more functions to create a new function. This concept is crucial in various fields, including calculus, algebra, and analysis.

Q6: Can I use the composition of functions to solve equations?

A6: Yes, you can use the composition of functions to solve equations. By composing two functions, you can create a new function that can be used to solve equations.

Q7: How do I find the inverse of a function?

A7: To find the inverse of a function, you need to swap the x and y variables and then solve for y.

Q8: What is the difference between a function and its inverse?

A8: The difference between a function and its inverse is the order in which the input and output values are swapped. A function takes an input value and produces an output value, while its inverse takes the output value and produces the input value.

Q9: Can I use the composition of functions to find the inverse of a function?

A9: Yes, you can use the composition of functions to find the inverse of a function. By composing the function with its inverse, you can create a new function that is equal to the identity function.

Q10: What are some real-world applications of composition of functions?

A10: The composition of functions has many real-world applications, including:

• Modeling population growth and decline
• Analyzing financial data
• Solving optimization problems
• Creating algorithms for computer science

Conclusion

In this article, we explored the composition of functions, including how to find the composition of two functions, how to determine if two functions are inverse functions, and how to use the composition of functions to solve equations. We also discussed the significance of composition of functions and its real-world applications.

Final Answer

To summarize, the final answers are:

• $(F \circ G)(x) = x$
• $(G \circ F)(x) = x$
• $F(x)$ and $G(x)$ are inverse functions.
• The composition of functions is a fundamental concept in mathematics that allows us to combine two or more functions to create a new function.
• The composition of functions has many real-world applications, including modeling population growth and decline, analyzing financial data, solving optimization problems, and creating algorithms for computer science.