https://file.aiquickdraw.com/mathbot/file/172736271231846orcq80.jpg

We are given the functions:

- \( f(x) = x + 1 \)
- \( g(x) = (x + 1)(3x - 4) \)

The rational function is defined as:

\[
m(x) = \frac{f(x)}{g(x)} = \frac{x + 1}{(x + 1)(3x - 4)}
\]

To analyze holes in the graph of \( m(x) \), let's simplify the expression for \( m(x) \):

\[
m(x) = \frac{x + 1}{(x + 1)(3x - 4)}
\]

We see that \( x + 1 \) cancels out from both the numerator and denominator, but we must be careful when \( x = -1 \). Canceling introduces a potential hole at \( x = -1 \), because the function is undefined there even though the cancellation might suggest otherwise.

After canceling, we get:

\[
m(x) = \frac{1}{3x - 4}
\]

Now, let's analyze:

- The function \( m(x) = \frac{1}{3x - 4} \) has a vertical asymptote where the denominator is 0, which occurs at \( 3x - 4 = 0 \) or \( x = \frac{4}{3} \).
- There is a hole at \( x = -1 \) because \( g(x) = 0 \) at \( x = -1 \), but it was canceled out during the simplification process.

Thus, the graph of \( m(x) \) has a hole at \( x = -1 \) and a vertical asymptote at \( x = \frac{4}{3} \).

The correct answer is **B**: "The graph of \( m \) has holes at \( x = -1 \) and \( x = \frac{4}{3} \)".

Let me know if you'd like more details or have any questions!

### Follow-up questions:
1. What are vertical asymptotes and how do they differ from holes?
2. How do we identify holes in rational functions?
3. How do cancellations affect the graph of a rational function?
4. What does it mean for a function to be undefined at a certain point?
5. Can a rational function have both holes and vertical asymptotes at the same point?

**Tip**: When simplifying rational functions, always check for cancellations that might introduce holes in the graph.

The function f is given by f(x) = x + 1, and the function g is given by g(x) = (x + 1)(3x - 4). Consider the rational function m defined by m(x) = f(x)/g(x). Which of the following is true about holes in the graph of m in the xy-plane?

Learn how to analyze the rational function m(x) = (x + 1) / [(x + 1)(3x - 4)] to identify holes at x = -1 and a vertical asymptote at x = 4/3. This problem is suitable for Grades 10-12, focusing on rational functions, cancellations, and graph analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation