Question
Let \( f(x)=\frac{x}{x-7} \) and \( g(x)=\frac{4}{x} \). Find the following functions. Simplify your answers. \( f(g(x))=\frac{4}{-7 x} \) \( g(f(x))= \)
Ask by Gough Logan. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( g(f(x)) = \frac{4x - 28}{x} \)
Solution
To find \( g(f(x)) \), we'll perform the composition of the two functions \( f \) and \( g \).
Given:
\[
f(x) = \frac{x}{x - 7} \quad \text{and} \quad g(x) = \frac{4}{x}
\]
**Step-by-Step Calculation:**
1. **Compose the Functions:**
\[
g(f(x)) = g\left( \frac{x}{x - 7} \right)
\]
2. **Substitute \( f(x) \) into \( g(x) \):**
\[
g\left( \frac{x}{x - 7} \right) = \frac{4}{\frac{x}{x - 7}}
\]
3. **Simplify the Expression:**
\[
\frac{4}{\frac{x}{x - 7}} = 4 \times \frac{x - 7}{x} = \frac{4(x - 7)}{x}
\]
4. **Expand the Numerator (Optional):**
\[
\frac{4(x - 7)}{x} = \frac{4x - 28}{x}
\]
**Final Simplified Form:**
\[
g(f(x)) = \frac{4(x - 7)}{x} = \frac{4x - 28}{x}
\]
**Summary:**
\[
\boxed{g(f(x)) = \frac{4x - 28}{x}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find \( g(f(x)) \), we start with the expressions for \( f(x) \) and \( g(x) \). We have: \[ f(x) = \frac{x}{x - 7} \] Now, substituting \( f(x) \) into \( g(x) \): \[ g(f(x)) = g\left(\frac{x}{x - 7}\right) = \frac{4}{\frac{x}{x - 7}} \] To simplify this, we can rewrite it: \[ g(f(x)) = 4 \cdot \frac{x - 7}{x} = \frac{4(x - 7)}{x} \] Now, expand and simplify: \[ g(f(x)) = \frac{4x - 28}{x} \] Thus, the result is: \[ g(f(x)) = 4 - \frac{28}{x} \]
