Find $f(x$\] And $g(x$\] Such That $h(x)=(f \circ G)(x$\], Where $h(x)=(4x+5)^8$.Choose The Correct Pair Of Functions:A. $f(x)=4x+5$, $g(x)=x^8$B. $f(x)=\sqrt[8]{x}$,

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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. 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 $f(g(x))$. In this article, we will explore how to find the composition of two functions, given the composite function $h(x) = (4x+5)^8$. Our goal is to determine the correct pair of functions $f(x)$ and $g(x)$ such that $h(x) = (f \circ g)(x)$.

Understanding the Composite Function

The composite function $h(x) = (4x+5)^8$ is a function of the form $(f \circ g)(x)$. To find the composition of $f$ and $g$, we need to understand the structure of the composite function. In this case, the composite function is a power function with a base of $(4x+5)$ and an exponent of $8$. This suggests that the function $g(x)$ is likely to be a linear function, while the function $f(x)$ is likely to be a power function.

Analyzing the Options

We are given two options for the pair of functions $f(x)$ and $g(x)$:

A. $f(x) = 4x+5$, $g(x) = x^8$ B. $f(x) = \sqrt[8]{x}$, $g(x) = x^8$

To determine the correct pair of functions, we need to analyze each option and see if it satisfies the condition $h(x) = (f \circ g)(x)$.

Option A: $f(x) = 4x+5$, $g(x) = x^8$

Let's start by analyzing option A. If we substitute $g(x) = x^8$ into the function $f(x) = 4x+5$, we get:

$f(g(x)) = f(x^8) = 4x^8 + 5$

This is not equal to the composite function $h(x) = (4x+5)^8$. Therefore, option A is not the correct pair of functions.

Option B: $f(x) = \sqrt[8]{x}$, $g(x) = x^8$

Now, let's analyze option B. If we substitute $g(x) = x^8$ into the function $f(x) = \sqrt[8]{x}$, we get:

$f(g(x)) = f(x^8) = \sqrt[8]{x^8} = x$

This is not equal to the composite function $h(x) = (4x+5)^8$. However, we can simplify the expression $f(g(x))$ by using the property of exponents:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the expression $f(g(x))$ as:

$f(g(x)) = \sqrt[8]{x^8} = (x^8)^{\frac{1}{8}} = x^1 = x$

This is still not equal to the composite function $h(x) = (4x+5)^8$. However, we can rewrite the

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Introduction

In our previous article, we explored how to find the composition of two functions, given the composite function $h(x) = (4x+5)^8$. We analyzed two options for the pair of functions $f(x)$ and $g(x)$ and determined that option B, $f(x) = \sqrt[8]{x}$, $g(x) = x^8$, is not the correct pair of functions. However, we can still use this option to answer some common questions about the composition of functions.

Q: What is the composition of functions?

A: The composition of functions is a way 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 $f(g(x))$.

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

A: To find the composition of two functions, you need to substitute the second function into the first function. For example, if we want to find the composition of $f(x) = \sqrt[8]{x}$ and $g(x) = x^8$, we would substitute $g(x)$ into $f(x)$ to get:

$f(g(x)) = f(x^8) = \sqrt[8]{x^8} = x$

Q: What is the difference between the composition of functions and the product of functions?

A: The composition of functions and the product of functions are two different mathematical operations. The composition of functions is a way of combining two or more functions to create a new function, while the product of functions is a way of multiplying two or more functions together.

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

A: Yes, the composition of functions can be used to solve equations. For example, if we have an equation of the form $f(g(x)) = h(x)$, we can use the composition of functions to solve for $x$.

Q: What are some common applications of the composition of functions?

A: The composition of functions has many common applications in mathematics and science. Some examples include:

• Modeling population growth and decay
• Analyzing the behavior of complex systems
• Solving equations and inequalities
• Finding the inverse of a function

Q: How do I know if a function is a composition of functions?

A: To determine if a function is a composition of functions, you need to look for the presence of parentheses or other grouping symbols. If a function has parentheses or other grouping symbols, it may be a composition of functions.

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

A: Yes, the composition of functions can be used to find the inverse of a function. If we have a function $f(x)$ and we want to find its inverse, we can use the composition of functions to find $f^{-1}(x)$.

Conclusion

In this article, we explored some common questions about the composition of functions. We discussed how to find the composition of two functions, the difference between the composition of functions and the product of functions, and some common applications of the composition of functions. We also provided some tips for determining if a function is a composition of functions and for using the composition of functions to find the inverse of a function.