https://file.aiquickdraw.com/mathbot/file/1725398392447xobxchlt.jpg

We are tasked with finding any holes in the function given as:

\[
f(x) = \frac{x^2 - 25}{3x + 15}
\]

### Step 1: Factor the numerator and denominator
First, let's factor the numerator and the denominator, if possible.

- The numerator is \( x^2 - 25 \), which is a difference of squares and can be factored as:
  \[
  x^2 - 25 = (x - 5)(x + 5)
  \]

- The denominator is \( 3x + 15 \), which can be factored by taking out the greatest common factor (GCF) of 3:
  \[
  3x + 15 = 3(x + 5)
  \]

Thus, the function becomes:
\[
f(x) = \frac{(x - 5)(x + 5)}{3(x + 5)}
\]

### Step 2: Simplify the expression
We notice that the factor \( (x + 5) \) appears in both the numerator and the denominator. Therefore, we can cancel the common factor, but we must account for any restrictions.

The simplified function is:
\[
f(x) = \frac{x - 5}{3}
\]
However, we must remember that \( x + 5 = 0 \) leads to a restriction. Specifically, \( x = -5 \) causes the original denominator to be zero. Even though the factor was canceled, this means there is a **hole** in the function at \( x = -5 \).

### Step 3: Determine the coordinate of the hole
To find the y-coordinate of the hole, substitute \( x = -5 \) into the simplified function:
\[
f(-5) = \frac{-5 - 5}{3} = \frac{-10}{3}
\]

Thus, the hole occurs at the point:
\[
(-5, -\frac{10}{3})
\]

### Final Answer
The function has a hole at the coordinate point \( (-5, -\frac{10}{3}) \).

---

Would you like further details or clarification? Here are some related questions for further practice:

1. How would you find vertical asymptotes in a rational function?
2. What are the steps to determine if a function has any horizontal asymptotes?
3. Can a function have more than one hole? If so, how do you find them?
4. What is the difference between a hole and a vertical asymptote?
5. How does canceling factors affect the domain of a function?

**Tip:** Always check for common factors between the numerator and denominator before simplifying rational functions to identify any holes in the function.

Find all holes of the following function. If a hole exists, write your answer as a coordinate point in simplest form. f(x) = (x^2 - 25) / (3x + 15)

Explore how to identify holes in the rational function f(x) = (x^2 - 25) / (3x + 15) by factoring and simplifying the expression. This detailed guide covers key algebraic concepts such as factoring differences of squares and identifying restrictions in the domain. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation