https://file.aiquickdraw.com/mathbot/file/1727470350394otr413cb.jpg

Let's analyze the function given in the image step-by-step:

### Function:
\[
f(x) = \frac{x^2 + 10x + 25}{x^2 + 11x + 30} = \frac{(x+5)^2}{(x+5)(x+6)}
\]

### Step 1: Simplify the function
We can simplify the function by canceling out the common factor of \((x+5)\):
\[
f(x) = \frac{(x+5)}{(x+6)}
\]
However, note that \(x = -5\) is a removable discontinuity (hole), as it would have been undefined in the original function before simplification.

### Step 2: Find the hole
The hole occurs at \(x = -5\). To find the \(y\)-coordinate of the hole, substitute \(x = -5\) into the simplified function:
\[
f(x) = \frac{(x+5)}{(x+6)}
\]
At \(x = -5\), the function becomes:
\[
f(-5) = \frac{-5+5}{-5+6} = \frac{0}{1} = 0
\]
Thus, the hole is at \((-5, 0)\).

### Step 3: Find the vertical asymptote
The vertical asymptote occurs where the denominator of the simplified function is zero. Setting \(x+6 = 0\) gives:
\[
x = -6
\]
Thus, the vertical asymptote is at \(x = -6\).

### Step 4: Find the horizontal asymptote
To find the horizontal asymptote, examine the degrees of the numerator and denominator in the simplified function:
\[
f(x) = \frac{x+5}{x+6}
\]
Both the numerator and denominator are of degree 1, so the horizontal asymptote is found by dividing the leading coefficients. Since the leading coefficients of both the numerator and denominator are 1, the horizontal asymptote is:
\[
y = \frac{1}{1} = 1
\]

### Final answers:
- **Hole**: \((-5, 0)\)
- **Vertical asymptote**: \(x = -6\)
- **Horizontal asymptote**: \(y = 1\)

Do you want further details or have any questions? Here are 5 related questions to deepen your understanding:

1. How do you distinguish between a hole and a vertical asymptote?
2. What are the steps to simplify a rational function?
3. How do you find the horizontal asymptote for rational functions with different degrees?
4. Why is there a hole at \(x = -5\) but no vertical asymptote there?
5. Can a function have more than one vertical asymptote?

**Tip**: Always check for common factors in both the numerator and the denominator before attempting to find holes and asymptotes!

Identify any holes and asymptotes for the function f(x) = (x^2 + 10x + 25) / (x^2 + 11x + 30) = ((x+5)^2) / ((x+5)(x+6)).

Learn how to find holes and asymptotes for the rational function f(x) = (x^2 + 10x + 25) / (x^2 + 11x + 30). The function simplifies, revealing both vertical and horizontal asymptotes and a hole at a removable discontinuity. Ideal for high school math students studying rational functions and asymptote behavior.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation