https://file.aiquickdraw.com/mathbot/file/1728940572006hngeb6xj.jpg

Let's work through the problem shown in the image. 

We are given:
- \( f(x) = \frac{1}{x - 2} \)
- \( g(x) = 4x + 7 \)

We need to find:
1. \( (f \circ g)(3) \)
2. \( (f \circ g)(x) \)

### Solution for \( (f \circ g)(x) \)

The composition \( (f \circ g)(x) \) means \( f(g(x)) \). We first substitute \( g(x) \) into \( f(x) \):

\[
(f \circ g)(x) = f(g(x)) = f(4x + 7)
\]

Substituting \( 4x + 7 \) into \( f(x) = \frac{1}{x - 2} \):

\[
f(4x + 7) = \frac{1}{(4x + 7) - 2} = \frac{1}{4x + 5}
\]

So, \( (f \circ g)(x) = \frac{1}{4x + 5} \).

### Solution for \( (f \circ g)(3) \)

Now, let's calculate \( (f \circ g)(3) \):

1. First, find \( g(3) \):

\[
g(3) = 4(3) + 7 = 12 + 7 = 19
\]

2. Next, substitute 19 into \( f(x) \):

\[
f(19) = \frac{1}{19 - 2} = \frac{1}{17}
\]

So, \( (f \circ g)(3) = \frac{1}{17} \).

### Summary

- \( (f \circ g)(3) = \frac{1}{17} \)
- \( (f \circ g)(x) = \frac{1}{4x + 5} \)

Would you like any further explanation or details on how the function composition works?

Here are 5 related questions for further exploration:
1. What is the domain of \( (f \circ g)(x) \)?
2. How would you find \( (g \circ f)(x) \) using the given functions?
3. What happens to \( (f \circ g)(x) \) if \( f(x) \) or \( g(x) \) is modified?
4. How can you determine whether \( (f \circ g)(x) \) has any vertical asymptotes?
5. What is the inverse of \( (f \circ g)(x) \), if it exists?

**Tip:** Remember that the domain of a composition function is determined by the domain of \( g(x) \) and any additional restrictions imposed by \( f(x) \) after substituting \( g(x) \).

Let f(x) = 1 / (x - 2) and g(x) = 4x + 7. Find (f ∘ g)(3) and (f ∘ g)(x).

Explore the composition of two functions, f(x) = 1 / (x - 2) and g(x) = 4x + 7. Learn how to calculate (f ∘ g)(3) and (f ∘ g)(x) with detailed steps, including substitution and simplifying rational functions. Ideal for students in Grades 10-12.

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