Let $f(x)=\sqrt{\frac{x^2+2x-15}{x-2}}$. Write The Domain Of $f$ As An Interval.$\square$

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with rational functions, it's essential to consider the restrictions on the domain caused by division by zero and the presence of square roots. In this article, we will explore the domain of a given rational function, $f(x)=\sqrt{\frac{x^2+2x-15}{x-2}}$, and express it as an interval.

Understanding the Function

The given function is a rational function, which means it is the ratio of two polynomials. The numerator is $x^2+2x-15$, and the denominator is $x-2$. To find the domain of this function, we need to consider the restrictions on the input values that make the function undefined.

Restrictions on the Domain

There are two main restrictions on the domain of the function:

1. Division by Zero: The function is undefined when the denominator is equal to zero. In this case, the denominator is $x-2$, so the function is undefined when $x=2$.
2. Square Root: The function involves a square root, which means the radicand (the expression inside the square root) must be non-negative. In this case, the radicand is $\frac{x^2+2x-15}{x-2}$.

Finding the Domain

To find the domain of the function, we need to consider the restrictions on the input values that make the function undefined. We will start by finding the values of $x$ that make the denominator equal to zero.

Denominator Equal to Zero

The denominator is equal to zero when $x-2=0$. Solving for $x$, we get:

$x-2=0$ $x=2$

So, the function is undefined when $x=2$.

Radicand Non-Negative

The radicand is non-negative when $\frac{x^2+2x-15}{x-2} \geq 0$. To find the values of $x$ that satisfy this inequality, we can start by factoring the numerator:

$x^2+2x-15 = (x+5)(x-3)$

So, the radicand can be written as:

$\frac{(x+5)(x-3)}{x-2}$

To find the values of $x$ that make the radicand non-negative, we can consider the signs of the factors.

Sign Analysis

We will analyze the signs of the factors $(x+5)$, $(x-3)$, and $(x-2)$ to determine when the radicand is non-negative.

• $(x+5)$: This factor is positive when $x \geq -5$ and negative when $x < -5$.
• $(x-3)$: This factor is positive when $x > 3$ and negative when $x \leq 3$.
• $(x-2)$: This factor is positive when $x > 2$ and negative when $x \leq 2$.

Combining the Signs

To find the values of $x$ that make the radicand non-negative, we need to combine the signs of the factors.

• $(x+5)$ and $(x-3)$ are both positive when $x > 3$.
• $(x+5)$ and $(x-3)$ are both negative when $x < -5$.
• $(x+5)$ is positive and $(x-3)$ is negative when $-5 < x \leq 3$.
• $(x+5)$ is negative and $(x-3)$ is positive when $x > 3$.

Radicand Non-Negative

The radicand is non-negative when $(x+5)(x-3) \geq 0$. This occurs when $x \leq -5$ or $x \geq 3$.

Domain of the Function

The domain of the function is the set of all possible input values for which the function is defined. Based on the restrictions on the domain, we can conclude that the function is undefined when $x=2$. The function is also undefined when the radicand is negative, which occurs when $-5 < x < 3$.

Conclusion

In conclusion, the domain of the function $f(x)=\sqrt{\frac{x^2+2x-15}{x-2}}$ is the set of all real numbers except $x=2$ and $-5 < x < 3$. This can be expressed as the interval $(-\infty, -5] \cup (3, \infty) \cup \{2\}$.

Final Answer

The final answer is $\boxed{(-\infty, -5] \cup (3, \infty) \cup \{2\}}$.