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We are given two functions:

1. \( f(x) = \frac{5x - 1}{3x + 4} \)
2. \( g(x) = \frac{5}{2x} \)

The problem asks us to find \( (f \circ g)(x) \), which is the composition of the functions \( f \) and \( g \). This means we need to substitute \( g(x) \) into \( f(x) \).

### Step-by-step solution:

To find \( (f \circ g)(x) \), we substitute \( g(x) = \frac{5}{2x} \) into \( f(x) \):

\[
(f \circ g)(x) = f(g(x)) = f\left( \frac{5}{2x} \right)
\]

Now, substitute \( \frac{5}{2x} \) for \( x \) in the expression for \( f(x) \):

\[
f\left( \frac{5}{2x} \right) = \frac{5\left(\frac{5}{2x}\right) - 1}{3\left(\frac{5}{2x}\right) + 4}
\]

Simplify both the numerator and denominator:

#### Numerator:

\[
5\left( \frac{5}{2x} \right) = \frac{25}{2x}
\]
\[
\frac{25}{2x} - 1 = \frac{25}{2x} - \frac{2x}{2x} = \frac{25 - 2x}{2x}
\]

#### Denominator:

\[
3\left( \frac{5}{2x} \right) = \frac{15}{2x}
\]
\[
\frac{15}{2x} + 4 = \frac{15}{2x} + \frac{8x}{2x} = \frac{15 + 8x}{2x}
\]

Thus, the composition \( (f \circ g)(x) \) becomes:

\[
(f \circ g)(x) = \frac{\frac{25 - 2x}{2x}}{\frac{15 + 8x}{2x}}
\]

Since both the numerator and denominator are divided by \( 2x \), these terms cancel out:

\[
(f \circ g)(x) = \frac{25 - 2x}{15 + 8x}
\]

### Final Answer:

\[
(f \circ g)(x) = \frac{25 - 2x}{15 + 8x}
\]

Would you like further explanation, or have any questions?

Here are 5 related questions to consider:

1. What is the domain of the function \( (f \circ g)(x) \)?
2. How would you compute \( (g \circ f)(x) \) in this case?
3. How can you verify the result of \( (f \circ g)(x) \) by plugging in specific values of \( x \)?
4. What happens to \( (f \circ g)(x) \) as \( x \) approaches 0?
5. How would you graph the composition \( (f \circ g)(x) \)?

**Tip:** When simplifying complex fractions, always look for common factors or terms that can cancel out to simplify the expression!

Jika f(x) = (5x - 1) / (3x + 4) dan g(x) = 5 / (2x), maka (f o g)(x) adalah ...

Learn how to solve the function composition problem involving f(x) = (5x - 1) / (3x + 4) and g(x) = 5 / (2x). Follow the step-by-step process of substituting g(x) into f(x) and simplifying the expression. Suitable for students in Grades 10-12.

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