To find the domain and range of the function $y = \frac{x^2 - 25}{x + 5}$, let's go through the steps:

1. Simplify the function

First, notice that the numerator $x^2 - 25$ can be factored as a difference of squares: $x^2 - 25 = (x - 5)(x + 5)$ So the function becomes: $y = \frac{(x - 5)(x + 5)}{x + 5}$ Now, we can cancel out the $(x + 5)$ terms, but we need to remember that we can't divide by zero, so $x + 5 \neq 0$, which means $x \neq -5$.

The simplified function is: $y = x - 5 \quad \text{for} \quad x \neq -5$

2. Domain

The domain refers to all the possible values that $x$ can take for which the function is defined.

• The function is defined for all real numbers except $x = -5$, because at $x = -5$, the denominator would be zero, which makes the function undefined.

So, the domain is: $\boxed{(-\infty, -5) \cup (-5, \infty)}$ This means $x$ can take any real number except $x = -5$.

3. Range

The range refers to all possible values of $y$ that the function can output.

• The simplified form of the function is $y = x - 5$, which is a linear function with a slope of 1 and a y-intercept of $-5$.
• Since this is a linear equation with no restrictions, $y$ can take any real number, but we need to check what happens when $x = -5$.

At $x = -5$, the original function is undefined. Substituting $x = -5$ into the simplified equation $y = x - 5$, we get: $y = -5 - 5 = -10$ So, while the function is undefined at $x = -5$, the output would have been $y = -10$ if it were defined. Therefore, $y = -10$ is not included in the range.

Thus, the range is: $\boxed{(-\infty, -10) \cup (-10, \infty)}$ This means that $y$ can take any real number except $y = -10$.

Explanation:

• Domain: The function is defined for all numbers except $x = -5$ because dividing by zero is undefined. So, the domain is all real numbers except $x = -5$.
• Range: The function outputs all numbers except $y = -10$ because at $x = -5$, the function is undefined, and the corresponding $y$-value would be $-10$.

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Would you like more details or have any questions?

5 Related Questions:

1. How do you find the vertical asymptotes of a rational function?
2. Can the domain or range of a function ever be restricted by other factors?
3. What happens to the function $y = x - 5$ as $x \to \infty$?
4. How can you visualize the graph of a rational function like this one?
5. What is the difference between a hole in the graph and an asymptote?

Tip: Always check for undefined values in rational functions by setting the denominator equal to zero.