To determine the range of the function $f(x) = \frac{5}{2x-1} + 4$, let us analyze the given function step by step.

Step 1: Identify any restrictions

The function $\frac{5}{2x-1}$ is undefined when the denominator $2x-1 = 0$, which happens at $x = \frac{1}{2}$. Hence, $f(x)$ is not defined at $x = \frac{1}{2}$, but this does not directly affect the range of $f(x)$.

Step 2: Rewrite the function for simplicity

Rewrite $f(x)$ as: $f(x) = \frac{5}{2x-1} + 4.$

The term $\frac{5}{2x-1}$ can take all real values except zero, because $2x-1 \neq 0$. Adding 4 to this expression shifts the range of $\frac{5}{2x-1}$ upward by 4.

Step 3: Analyze the shifted range

The value $f(x)$ will never be equal to 4 because $\frac{5}{2x-1}$ cannot be zero. Hence, the range of $f(x)$ is all real numbers except $4$.

Final Answer:

The range $R_f = \mathbb{R} - \{4\}$, corresponding to option B.

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Related Questions:

1. How do you find the domain of $f(x)$ for similar rational functions?
2. What happens to the range if a constant other than 4 is added?
3. How would you verify the range of $f(x)$ using calculus (derivatives)?
4. How is the asymptote of $f(x)$ determined?
5. Can the range include 4 if $f(x)$ is slightly modified?

Tip:

Always check for horizontal asymptotes and discontinuities when analyzing the range of rational functions.