Question

For f left parenthesis x right parenthesis equals StartFraction 7 Over x plus 8 EndFraction
and g left parenthesis x right parenthesis equals StartFraction 7 Over x EndFraction
​, find
a.  left parenthesis f circle g right parenthesis left parenthesis x right parenthesis​;     b.  the domain of f circle g
Question content area bottom
Part 1
a.  left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
  
enter your response here
​(Simplify your​ answer.)
Part 2
b.  What is the domain of f circle g​?
The domain is
  
enter your response here.
​(Simplify your answer. Type your answer in interval notation. Use integers or fractions for any numbers in the​ expression.)

Answer 1

Let's find \( (f \circ g)(x) \) and the domain of \( f \circ g \).

### Part 1: Finding \( (f \circ g)(x) \)

We have:
- \( f(x) = \frac{7}{x + 8} \)
- \( g(x) = \frac{7}{x} \)

To find \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \):

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

Now, replace \( x \) in \( f(x) \) with \( g(x) \):

\[
f\left(\frac{7}{x}\right) = \frac{7}{\frac{7}{x} + 8}
\]

Now, simplify the denominator:

\[
\frac{7}{x} + 8 = \frac{7 + 8x}{x}
\]

So, we have:

\[
f\left(\frac{7}{x}\right) = \frac{7}{\frac{7 + 8x}{x}} = \frac{7x}{7 + 8x}
\]

Thus,

\[
(f \circ g)(x) = \frac{7x}{7 + 8x}
\]

### Part 2: Finding the Domain of \( f \circ g \)

To determine the domain of \( (f \circ g)(x) = \frac{7x}{7 + 8x} \), we need to identify values of \( x \) that make the denominator zero:

Set the denominator equal to zero and solve:

\[
7 + 8x = 0
\]

\[
8x = -7 \quad \Rightarrow \quad x = -\frac{7}{8}
\]

The function \( g(x) = \frac{7}{x} \) is also undefined when \( x = 0 \), as the denominator cannot be zero.

Thus, the domain of \( (f \circ g)(x) \) is all real numbers except \( x = -\frac{7}{8} \) and \( x = 0 \).

### In Interval Notation

The domain in interval notation can be expressed as:

\[
(-\infty, -\frac{7}{8}) \cup (-\frac{7}{8}, 0) \cup (0, \infty)
\]

### Final Answers

a. \( (f \circ g)(x) = \frac{7x}{7 + 8x} \)

b. The domain is \( (-\infty, -\frac{7}{8}) \cup (-\frac{7}{8}, 0) \cup (0, \infty) \)