To simplify the function \( f(x) = \frac{4x - 1}{2x + 3} \), follow these steps:

1. **Identify common factors**: First, check if there's a common factor in both the numerator and the denominator that can be factored out. In this case, the expressions \( 4x - 1 \) and \( 2x + 3 \) do not have a common factor.

2. **Factorization (if possible)**: Check if either the numerator or denominator can be factored further. Here, both are already in their simplest linear forms.

3. **Check for simplification**: Since there are no common factors and both expressions are linear, the function is already in its simplest form.

So, the simplified form of the function is:

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

This is as simplified as the function can be.

Would you like more details or have any questions? Here are five related questions that might interest you:

1. How do you determine when a rational function can be simplified?
2. What is the domain of the function \( f(x) = \frac{4x - 1}{2x + 3} \)?
3. How do you find the vertical asymptotes of a rational function?
4. What are the intercepts of the function \( f(x) = \frac{4x - 1}{2x + 3} \)?
5. How can you find the horizontal asymptote of a rational function?

**Tip:** Always check for common factors between the numerator and the denominator before concluding that a rational function is fully simplified.

Learn how to simplify the rational function f(x) = (4x - 1) / (2x + 3) with a step-by-step guide. This problem covers key algebraic concepts, including rational functions and simplification techniques. Ideal for students in Grades 9-12.

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