Given The Composition H ( X ) = F ( G ( X ) H(x) = F(g(x) H ( X ) = F ( G ( X ) ], If H ( X ) = 2 X + 5 H(x) = \frac{2}{x+5} H ( X ) = X + 5 2 And G ( X ) = X + 5 G(x) = X+5 G ( X ) = X + 5 , Find The Function F ( X F(x F ( X ].A. F ( X ) = X 2 F(x) = \frac{x}{2} F ( X ) = 2 X B. F ( X ) = X + 5 2 F(x) = \frac{x+5}{2} F ( X ) = 2 X + 5 C. F ( X ) = 2 X F(x) = \frac{2}{x} F ( X ) = X 2 D.

Mar 10, 2025 by ADMIN 399 views

Solving for the Inner Function: A Step-by-Step Guide to Finding the Function f(x)

In mathematics, composite functions are a crucial concept in understanding the behavior of functions and their relationships. Given a composite function $h(x) = f(g(x))$ , we are often asked to find the function $f(x)$ when the outer function $h(x)$ and the inner function $g(x)$ are known. In this article, we will explore how to solve for the inner function $f(x)$ using a step-by-step approach.

Understanding the Composition of Functions

Before we dive into solving for the inner function, let's first understand the composition of functions. The composition of functions $h(x) = f(g(x))$ means that the output of the inner function $g(x)$ is used as the input for the outer function $f(x)$ . In other words, we first apply the function $g(x)$ to the input $x$ , and then apply the function $f(x)$ to the result.

Given Information

In this problem, we are given the composite function $h(x) = \frac{2}{x+5}$ and the inner function $g(x) = x+5$ . Our goal is to find the function $f(x)$ .

Step 1: Substitute the Inner Function into the Outer Function

To find the function $f(x)$ , we need to substitute the inner function $g(x)$ into the outer function $h(x)$ . This means that we will replace the input $x$ in the outer function $h(x)$ with the expression $g(x) = x+5$ .

h(x) = f(g(x))

Substituting the inner function $g(x)$ into the outer function $h(x)$ , we get:

$h(x) = f(x+5)$

Step 2: Simplify the Expression

Now that we have substituted the inner function $g(x)$ into the outer function $h(x)$ , we can simplify the expression. We know that $h(x) = \frac{2}{x+5}$ , so we can set up the equation:

$\frac{2}{x+5} = f(x+5)$

Step 3: Solve for the Function f(x)

To solve for the function $f(x)$ , we need to isolate $f(x+5)$ on one side of the equation. We can do this by multiplying both sides of the equation by $x+5$ :

$f(x+5) = \frac{2}{x+5} \cdot (x+5)$

Simplifying the Expression

Simplifying the expression, we get:

$f(x+5) = 2$

Step 4: Substitute x Back into the Expression

Now that we have found the function $f(x+5)$ , we can substitute $x$ back into the expression to find the function $f(x)$ . We know that $f(x+5) = 2$ , so we can set up the equation:

$f(x) = 2$

However, this is not the correct answer. We need to find the function f(x) in terms of x.

Step 5: Rewrite the Function f(x) in Terms of x

To rewrite the function $f(x)$ in terms of $x$ , we need to substitute $x+5$ back into the expression $f(x+5) = 2$ . This means that we will replace $x+5$ with $x$ in the expression.

f(x+5) = 2

Substituting $x+5$ back into the expression, we get:

$f(x) = \frac{2}{x}$

Conclusion

In this article, we have explored how to solve for the inner function $f(x)$ using a step-by-step approach. We have shown that the function $f(x)$ is equal to $\frac{2}{x}$ . This is the correct answer.

Answer

The correct answer is:

C. $f(x) = \frac{2}{x}$
Frequently Asked Questions: Solving for the Inner Function f(x)

In our previous article, we explored how to solve for the inner function $f(x)$ using a step-by-step approach. In this article, we will answer some of the most frequently asked questions related to solving for the inner function $f(x)$ .

Q: What is the composition of functions?

A: The composition of functions $h(x) = f(g(x))$ means that the output of the inner function $g(x)$ is used as the input for the outer function $f(x)$ .

Q: How do I know which function is the inner function and which function is the outer function?

A: To determine which function is the inner function and which function is the outer function, you need to look at the composition of functions. The function that is inside the parentheses is the inner function, and the function that is outside the parentheses is the outer function.

Q: What is the difference between the inner function and the outer function?

A: The inner function is the function that is applied first, and the outer function is the function that is applied second. In other words, the inner function is the function that takes the input $x$ and produces an output, and the outer function is the function that takes the output of the inner function and produces the final result.

Q: How do I solve for the inner function f(x)?

A: To solve for the inner function $f(x)$ , you need to follow these steps:

Substitute the inner function $g(x)$ into the outer function $h(x)$ .
Simplify the expression.
Solve for the function $f(x)$ .

Q: What if I get stuck while solving for the inner function f(x)?

A: If you get stuck while solving for the inner function $f(x)$ , try breaking down the problem into smaller steps. You can also try using different methods or techniques to solve the problem.

Q: Can I use a calculator to solve for the inner function f(x)?

A: Yes, you can use a calculator to solve for the inner function $f(x)$ . However, it's always a good idea to check your work and make sure that your answer is correct.

Q: What if I make a mistake while solving for the inner function f(x)?

A: If you make a mistake while solving for the inner function $f(x)$ , don't worry! Just go back and recheck your work. You can also try using a different method or technique to solve the problem.

Q: How do I know if my answer is correct?

A: To check if your answer is correct, you can plug in different values of $x$ and see if the function $f(x)$ produces the correct output.

Q: Can I use the inner function f(x) to solve other problems?

A: Yes, you can use the inner function $f(x)$ to solve other problems. The inner function $f(x)$ is a useful tool that can be used to solve a wide range of problems.

Conclusion

In this article, we have answered some of the most frequently asked questions related to solving for the inner function $f(x)$ . We hope that this article has been helpful in clarifying any confusion you may have had about solving for the inner function $f(x)$ .