why is x-8/3x-16 listed as the correct answer when 𝑓(𝑥)=1𝑥+2f(x)=1x+2 and 𝑔(𝑥)=𝑥/𝑥−8 and im evaluating f(g(x))


Let's go step by step to evaluate \( f(g(x)) \) when \( f(x) = \frac{1}{x+2} \) and \( g(x) = \frac{x}{x-8} \).

### Step 1: Substitute \( g(x) \) into \( f(x) \)
To find \( f(g(x)) \), we substitute the expression for \( g(x) \) into \( f(x) \). This means wherever there is an \( x \) in the function \( f(x) \), we replace it with \( g(x) \).

\[
f(g(x)) = \frac{1}{g(x) + 2}
\]

### Step 2: Replace \( g(x) \) with \( \frac{x}{x-8} \)
Now, substitute the expression for \( g(x) \) into the equation for \( f(g(x)) \):

\[
f(g(x)) = \frac{1}{\frac{x}{x-8} + 2}
\]

### Step 3: Simplify the expression
We need to simplify the denominator. First, express the "2" as a fraction with a common denominator:

\[
f(g(x)) = \frac{1}{\frac{x}{x-8} + \frac{2(x-8)}{x-8}}
\]
\[
f(g(x)) = \frac{1}{\frac{x + 2(x-8)}{x-8}}
\]
Now simplify the numerator:

\[
x + 2(x-8) = x + 2x - 16 = 3x - 16
\]

So the expression becomes:

\[
f(g(x)) = \frac{1}{\frac{3x - 16}{x-8}}
\]

### Step 4: Simplify further
Now, simplify the overall expression:

\[
f(g(x)) = \frac{1}{\frac{3x - 16}{x-8}} = \frac{x - 8}{3x - 16}
\]

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

\[
f(g(x)) = \frac{x - 8}{3x - 16}
\]

This explains why the expression \( \frac{x - 8}{3x - 16} \) is listed as the correct answer.

Would you like any further clarification or more details?

### Related Questions:
1. How do you evaluate composite functions like \( f(g(x)) \) for more complex functions?
2. What are the typical domains for functions like \( f(x) = \frac{1}{x+2} \) and \( g(x) = \frac

why is x-8/3x-16 listed as the correct answer when f(x)=1/x+2 and g(x)=x/(x-8) and I'm evaluating f(g(x))

Learn how to evaluate the composite function f(g(x)) where f(x)=1/(x+2) and g(x)=x/(x-8). This detailed step-by-step solution explains the simplification process and algebraic manipulation involved in solving rational functions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation