Let f(x) = x / (x-7) and g(x) = 3 / x

Find the function of g(f(x))

To find the function \( g(f(x)) \), we need to substitute \( f(x) \) into the function \( g(x) \).

Given:
\[
f(x) = \frac{x}{x - 7}
\]
\[
g(x) = \frac{3}{x}
\]

We substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g\left(\frac{x}{x - 7}\right)
\]

Now, replace \( x \) in \( g(x) \) with \( \frac{x}{x - 7} \):

\[
g(f(x)) = \frac{3}{\frac{x}{x - 7}}
\]

Next, simplify the expression:

\[
g(f(x)) = \frac{3 \cdot (x - 7)}{x}
\]

Distribute the 3 in the numerator:

\[
g(f(x)) = \frac{3x - 21}{x}
\]

Simplify further:

\[
g(f(x)) = 3 - \frac{21}{x}
\]

So, the function \( g(f(x)) \) is:

\[
g(f(x)) = 3 - \frac{21}{x}
\]

Would you like more details or have any questions about this process?

Here are some related questions:

1. How do you find the domain of \( g(f(x)) \)?
2. What is \( f(g(x)) \)?
3. How do you determine the inverse of \( g(x) \)?
4. What is the value of \( g(f(7)) \)?
5. How would you graph \( g(f(x)) \)?

**Tip:** When working with composite functions, always check for simplifications that might make the function easier to understand or work with.

Explore how to find the composite function g(f(x)) using the functions f(x) = x / (x - 7) and g(x) = 3 / x. Step-by-step solution provided with detailed algebraic simplifications. Suitable for high school students studying functions and composite functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation