To find the range of the function

$y = \sqrt{\frac{4x + 1}{x - 1}},$

we need to determine the set of possible values for $y$ based on the function's behavior.

Step 1: Identify the domain

For the expression under the square root to be valid, the following condition must be satisfied:

$\frac{4x + 1}{x - 1} \geq 0.$

Additionally, the denominator $x - 1$ cannot be zero, so $x \neq 1$.

Step 2: Analyze the expression

Let's solve the inequality:

$\frac{4x + 1}{x - 1} \geq 0.$

We'll first find the points where the numerator and denominator are zero:

• The numerator $4x + 1 = 0$ gives $x = -\frac{1}{4}$.
• The denominator $x - 1 = 0$ gives $x = 1$.

Thus, the critical points are $x = -\frac{1}{4}$ and $x = 1$.

Now, we'll check the sign of $\frac{4x + 1}{x - 1}$ in the intervals formed by these critical points: $(-\infty, -\frac{1}{4})$, $(-\frac{1}{4}, 1)$, and $(1, \infty)$.

• For $x < -\frac{1}{4}$, both $4x + 1$ and $x - 1$ are negative, so $\frac{4x + 1}{x - 1} > 0$.
• For $-\frac{1}{4} < x < 1$, $4x + 1$ is positive and $x - 1$ is negative, so $\frac{4x + 1}{x - 1} < 0$.
• For $x > 1$, both $4x + 1$ and $x - 1$ are positive, so $\frac{4x + 1}{x - 1} > 0$.

Step 3: Determine where $\frac{4x + 1}{x - 1} \geq 0$

From the sign analysis, we conclude that the inequality holds when:

$x \in (-\infty, -\frac{1}{4}] \cup (1, \infty).$

Thus, the domain of the function is $x \in (-\infty, -\frac{1}{4}] \cup (1, \infty)$.

Step 4: Behavior of the function at the boundaries

We examine the behavior of the function near the boundaries of the domain:

• As $x \to -\frac{1}{4}^-$, $\frac{4x + 1}{x - 1} \to 0$, so $y = \sqrt{0} = 0$.
• As $x \to -\infty$, $\frac{4x + 1}{x - 1} \to 4$, so $y = \sqrt{4} = 2$.
• As $x \to 1^+$, $\frac{4x + 1}{x - 1} \to \infty$, so $y \to \infty$.
• As $x \to \infty$, $\frac{4x + 1}{x - 1} \to 4$, so $y = \sqrt{4} = 2$.

Step 5: Conclusion

The range of the function is:

$y \in [0, \infty).$

Summary

The range of the function $y = \sqrt{\frac{4x + 1}{x - 1}}$ is $[0, \infty)$.

Would you like further details on how the solution was derived, or is there another question related to this topic?

Here are 5 related questions you might find interesting:

1. How do you find the domain of a rational function with a square root?
2. What happens to the function when we replace the square root with a cube root?
3. How can we graph this function to visualize its behavior?
4. How does the range change if we modify the numerator or denominator of the function?
5. How can we find the range of a rational function with a square root without graphing?

Tip: Always consider both the behavior at critical points and asymptotic behavior to understand the range and domain of functions involving square roots.