For f left parenthesis x right parenthesis equals StartFraction 7 Over x plus 5 EndFraction
and g left parenthesis x right parenthesis equals StartFraction 3 Over x EndFraction
​, find
a.  left parenthesis f circle g right parenthesis left parenthesis x right parenthesis​;     b.  the domain of f circle g

Let's solve each part step by step.

### Given functions:
- \( f(x) = \frac{7}{x + 5} \)
- \( g(x) = \frac{3}{x} \)

### Part a: Find \( (f \circ g)(x) \)

The notation \( (f \circ g)(x) \) means the composition of \( f \) and \( g \), or \( f(g(x)) \). This means we substitute \( g(x) \) into \( f(x) \).

\[
(f \circ g)(x) = f(g(x)) = f\left(\frac{3}{x}\right)
\]

Now, substitute \( \frac{3}{x} \) into the expression for \( f(x) \):

\[
f\left(\frac{3}{x}\right) = \frac{7}{\frac{3}{x} + 5}
\]

To simplify, we combine the terms in the denominator:

\[
\frac{3}{x} + 5 = \frac{3 + 5x}{x}
\]

Now, the expression becomes:

\[
f\left(\frac{3}{x}\right) = \frac{7}{\frac{3 + 5x}{x}} = \frac{7x}{3 + 5x}
\]

Thus, the composition \( (f \circ g)(x) = \frac{7x}{3 + 5x} \).

### Part b: Find the domain of \( f \circ g \)

The domain of \( f \circ g \) is determined by two things:
1. The domain of \( g(x) \), which must exclude values that make the denominator zero.
2. The domain of \( f(g(x)) \), which must also exclude values that make the denominator zero after substituting \( g(x) \) into \( f(x) \).

#### Step 1: Domain of \( g(x) \)
The function \( g(x) = \frac{3}{x} \) is undefined when \( x = 0 \), so \( x \neq 0 \).

#### Step 2: Domain of \( f(g(x)) \)
We have \( f(g(x)) = \frac{7x}{3 + 5x} \), and this is undefined when the denominator equals zero. So, solve:

\[
3 + 5x = 0 \quad \Rightarrow \quad x = -\frac{3}{5}
\]

Thus, \( x = -\frac{3}{5} \) must also be excluded from the domain.

### Conclusion for the domain:
The domain of \( (f \circ g)(x) \) is all real numbers except \( x = 0 \) and \( x = -\frac{3}{5} \). In interval notation, this is:

\[
\boxed{(-\infty, -\frac{3}{5}) \cup (-\frac{3}{5}, 0) \cup (0, \infty)}
\]

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Would you like more details on any step? Here are some related questions to deepen understanding:

1. How do you find the composition of two functions in general?
2. What is the domain of \( f \) alone, and how does it compare to the domain of \( g \)?
3. How would the result change if \( g(x) \) had a different denominator?
4. Can you explain how to simplify rational expressions in function compositions?
5. What happens if \( g(x) \) is a more complex rational function?

**Tip:** When determining the domain of a composition, always check the domain restrictions of both the inside and outside functions.

For f(x) = 7 / (x + 5) and g(x) = 3 / x, find a. (f ∘ g)(x); b. the domain of (f ∘ g).

Learn how to find the composition of functions f(x) = 7 / (x + 5) and g(x) = 3 / x, and determine the domain of (f ∘ g)(x). This step-by-step guide covers key concepts like function composition and rational functions, suitable for students in Grades 10-12.

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