Fine the domain and range of y= (x²-25) over (x+5) then explain the answer like all numbers, greater or less than or whatsoever it is

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.

Find the domain and range of y= (x²-25) over (x+5) then explain the answer like all numbers, greater or less than or whatsoever it is.

Understand how to find the domain and range of the rational function y = (x²-25) / (x+5) with a step-by-step solution. This explanation includes algebraic concepts like rational functions, factoring, and identifying undefined points. Ideal for high school students in Grades 9-11.

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