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The problem in the image is asking to determine the domains of two functions \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{5x} \), as well as the domain of the composition of these functions \( (f \circ g)(x) \).

### Solution:

#### (a) Domain of \( f(x) = \frac{1}{x} \)
The domain of a function is the set of all input values \( x \) for which the function is defined. For \( f(x) = \frac{1}{x} \), the function is undefined when the denominator is zero. Therefore:
\[
x \neq 0
\]
The domain of \( f(x) \) is all real numbers except zero, which in interval notation is:
\[
(-\infty, 0) \cup (0, \infty)
\]

#### (b) Domain of \( g(x) = \frac{1}{5x} \)
Similarly, for \( g(x) = \frac{1}{5x} \), the denominator is \( 5x \), and the function is undefined when \( 5x = 0 \), or equivalently when \( x = 0 \). Thus, the domain of \( g(x) \) is:
\[
(-\infty, 0) \cup (0, \infty)
\]

#### (c) Domain of \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) = f(g(x)) \) means applying \( f \) to the output of \( g(x) \). First, \( g(x) = \frac{1}{5x} \), so:
\[
f(g(x)) = f\left(\frac{1}{5x}\right) = \frac{1}{\frac{1}{5x}} = 5x
\]
Now, \( f(g(x)) = 5x \) is a linear function, which is defined for all real numbers except when \( x = 0 \), since \( g(x) \) is undefined at \( x = 0 \). Therefore, the domain of \( f \circ g \) is also:
\[
(-\infty, 0) \cup (0, \infty)
\]

### Summary:
- (a) Domain of \( f \): \( (-\infty, 0) \cup (0, \infty) \)
- (b) Domain of \( g \): \( (-\infty, 0) \cup (0, \infty) \)
- (c) Domain of \( f \circ g \): \( (-\infty, 0) \cup (0, \infty) \)

Would you like further clarification or details on any part of the solution?

Here are 5 related questions to deepen your understanding:
1. How do you determine the domain of a function involving fractions?
2. What happens if you compose two functions with different domains?
3. How do you graph a function like \( f(x) = \frac{1}{x} \)?
4. What is the general rule for determining the domain of a composite function?
5. Can there be other restrictions on the domain besides the denominator being zero?

**Tip:** Always check for points where a function is undefined, such as divisions by zero or negative values under a square root, when finding the domain.

Determine the domains of the functions f(x) = 1/x and g(x) = 1/(5x), as well as the domain of the composition of these functions f ∘ g.

Explore how to determine the domains of the functions f(x) = 1/x and g(x) = 1/(5x), and the composite function f ∘ g. This problem covers algebraic concepts of rational functions and their domains. Suitable for students in Grades 10-12.

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