https://file.aiquickdraw.com/mathbot/file/1731551288722194v4r1f.jpg

Let's analyze the problem presented.

The function \( f(x) \) is defined as a piecewise function:

\[
f(x) = \begin{cases} 
5x - 4 & \text{if } x \leq 1 \\ 
-x + 4 & \text{if } x > 1 
\end{cases}
\]

The task involves graphing this function and determining if it is continuous.

### Steps to Determine Continuity at \( x = 1 \):

1. **Evaluate \( f(1) \) Using the First Case:** Since \( x = 1 \) is within the first piece, \( f(1) = 5(1) - 4 = 5 - 4 = 1 \).

2. **Calculate the Left-Hand Limit as \( x \to 1^- \):**
   Using \( f(x) = 5x - 4 \) as \( x \to 1^- \), we get:
   \[
   \lim_{{x \to 1^-}} f(x) = 5(1) - 4 = 1
   \]

3. **Calculate the Right-Hand Limit as \( x \to 1^+ \):**
   Using \( f(x) = -x + 4 \) as \( x \to 1^+ \), we get:
   \[
   \lim_{{x \to 1^+}} f(x) = -1 + 4 = 3
   \]

Since the left-hand limit (\(1\)) and the right-hand limit (\(3\)) are not equal, the function is **discontinuous at \( x = 1 \)**. This is because, for continuity at \( x = 1 \), we would need:
\[
\lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1)
\]

Thus, the answer to whether the function is continuous is **No**.

Would you like more details or have any questions?

Here are some related questions to consider:
1. How do we determine continuity for piecewise functions in general?
2. What is the importance of left-hand and right-hand limits in continuity?
3. How would the graph of \( f(x) \) look with the discontinuity at \( x = 1 \)?
4. Can we determine the type of discontinuity present here?
5. How would the continuity change if the pieces of \( f(x) \) were adjusted to make the limits match?

**Tip:** Always check both left-hand and right-hand limits when testing continuity at a point for piecewise functions.

Suppose that the function f is defined, for all real numbers, as follows: f(x) = { 5x - 4 if x ≤ 1 ; -x + 4 if x > 1 }. Graph the function f. Then determine whether or not the function is continuous.

Analyze the continuity of the piecewise function f(x) with definitions for x ≤ 1 and x > 1. This problem involves evaluating left and right limits and applying continuity conditions for functions at specific points. Suitable for high school 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