Let 𝑓(𝑥)=1𝑥+5
 and 𝑔(𝑥)=𝑥𝑥−9
 . Evaluate the following 𝑓(𝑔(𝑥))

We are asked to find \( f(g(x)) \), where \( f(x) = \frac{1}{x+5} \) and \( g(x) = \frac{x}{x-9} \).

To evaluate \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f\left(\frac{x}{x-9}\right)
\]
Since \( f(x) = \frac{1}{x+5} \), we replace \( x \) in \( f(x) \) with \( \frac{x}{x-9} \):

\[
f(g(x)) = \frac{1}{\left(\frac{x}{x-9}\right) + 5}
\]

Now, let's simplify the denominator:

\[
\left(\frac{x}{x-9}\right) + 5 = \frac{x}{x-9} + \frac{5(x-9)}{x-9} = \frac{x + 5(x-9)}{x-9}
\]
Simplifying further:
\[
x + 5(x-9) = x + 5x - 45 = 6x - 45
\]

So the expression becomes:

\[
f(g(x)) = \frac{1}{\frac{6x - 45}{x-9}} = \frac{x-9}{6x-45}
\]

Thus, the simplified expression for \( f(g(x)) \) is:

\[
f(g(x)) = \frac{x-9}{6x-45}
\]

Would you like a deeper explanation or any details on the steps? Here are some related questions you might consider:

1. What are the domain restrictions for \( f(g(x)) \)?
2. How would you evaluate \( g(f(x)) \)?
3. How do you simplify rational expressions?
4. What happens when \( x = 9 \) for the function \( g(x) \)?
5. How does function composition affect the domain and range?

**Tip:** Always check for any restrictions in the domain when dealing with rational functions, as division by zero can occur.

Let f(x)=1/(x+5) and g(x)=x/(x-9). Evaluate the following f(g(x)).

Learn how to evaluate the function composition f(g(x)) where f(x)=1/(x+5) and g(x)=x/(x-9). This detailed solution walks through the process of substituting g(x) into f(x) and simplifying rational expressions. Suitable for students in Grades 10-12.

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