Express the function h(x)=1/x+2 in the form f of g if g(x)=(x+2). find the function f(x)

To express the function \( h(x) = \frac{1}{x+2} \) in the form \( h(x) = f(g(x)) \), given that \( g(x) = x + 2 \), we follow these steps:

### Step-by-Step Breakdown:
1. **Understand the function form**: The function \( h(x) \) is to be expressed as a composition of two functions, \( f(x) \) and \( g(x) \), where:
   \[
   h(x) = f(g(x))
   \]
   and \( g(x) = x + 2 \).

2. **Substitute \( g(x) \) into \( h(x) \)**: 
   Since \( g(x) = x + 2 \), the expression \( h(x) = \frac{1}{x+2} \) can be written as:
   \[
   h(x) = \frac{1}{g(x)}
   \]

3. **Identify the function \( f(x) \)**: 
   From the equation above, we see that \( f(g(x)) = \frac{1}{g(x)} \). Therefore, the function \( f(x) \) is:
   \[
   f(x) = \frac{1}{x}
   \]

### Conclusion:
Thus, the function \( f(x) = \frac{1}{x} \).

Would you like to go over more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the domain of the function \( h(x) = \frac{1}{x+2} \)?
2. How would you find the inverse of the function \( f(x) = \frac{1}{x} \)?
3. What is the composition of \( f(g(x)) \) if \( g(x) = 2x \)?
4. How do you differentiate \( h(x) = \frac{1}{x+2} \)?
5. What transformations does the function \( f(x) = \frac{1}{x} \) undergo to become \( h(x) = \frac{1}{x+2} \)?

**Tip**: When composing functions, always check if any domain restrictions carry over from the inner function to the overall composition.

Express the function h(x)=1/x+2 in the form f of g if g(x)=(x+2). Find the function f(x).

Learn how to express the function h(x) = 1/(x+2) in the form f(g(x)) with a detailed step-by-step breakdown. This problem covers key algebraic concepts such as function composition and how to identify f(x) when g(x) is given. Suitable for students in Grades 9-11.

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