Graph t, locate all points of discontinuty, end discuss the behavior of t at these points. $$ I(x)=\left\{\begin{array}{ll} 1+x & \text { if } x

Question

Graph t, locate all points of discontinuty, end discuss the behavior of t at these points. $$ I(x)=\left\{\begin{array}{ll} 1+x & \text { if } x<1 \ 6-x & \text { if } x \geq 1 \end{array}\right. $$ Locate all points of discontinuity. Solect the conect choice below and, if necessary, fill in the answor box to complete your choice. A. The poirl(s) of d scontinuity is/are $$ \mathrm{x}= $$ $$ \square $$ (Use a comma to separate answers as needed.) B. There are no points of discontinuity Discuss the behavior of t at its point(s) of discontinuity. Find $$ \mathrm{f}(\mathrm{x}) $$ at any points of discontinuity. Select the corred choice below and, if necessary, fill in the answer hox to complete your choice A. At the poinl(s) of discontinuity, the value(s) of the function $$ f(x) $$ is/are 5 . (Use a comma to separate answers as needed.) B. There are no points of discontinuity.

UpStudy AI · Accepted Answer

Let's analyze the piecewise function $$ I(x) $$:

$$
I(x) = \begin{cases} 
1 + x & \text{if } x < 1 \
6 - x & \text{if } x \geq 1 
\end{cases}
$$

### 1. Locating Points of Discontinuity

To determine if there's a discontinuity at $$ x = 1 $$, we need to check the left-hand limit, the right-hand limit, and the function's value at $$ x = 1 $$.

- **Left-hand limit as $$ x $$ approaches 1:**
  $$
  \lim_{x \to 1^-} I(x) = 1 + 1 = 2
  $$

- **Right-hand limit as $$ x $$ approaches 1:**
  $$
  \lim_{x \to 1^+} I(x) = 6 - 1 = 5
  $$

- **Function value at $$ x = 1 $$:**
  $$
  I(1) = 6 - 1 = 5
  $$

Since the left-hand limit ($$ 2 $$) does not equal the right-hand limit ($$ 5 $$), there is a **point of discontinuity** at $$ x = 1 $$.

**Answer Choice for Discontinuity:**
- **A.** The point(s) of discontinuity is/are $$ \mathrm{x}= 1 $$

### 2. Behavior of $$ I(x) $$ at the Point of Discontinuity

At $$ x = 1 $$, the function value is $$ I(1) = 5 $$. This means that while the right-hand limit and the function's value at $$ x = 1 $$ are both 5, the left-hand limit is different, causing a jump discontinuity.

**Answer Choice for Function Behavior:**
- **A.** At the point(s) of discontinuity, the value(s) of the function $$ I(x) $$ is/are 5.

### Summary

- **Points of Discontinuity:** $$ x = 1 $$
- **Function Value at Discontinuity:** $$ I(1) = 5 $$

**Final Answers:**

1. **Locate all points of discontinuity.**
   - **A.** The point(s) of discontinuity is/are $$ x = 1 $$

2. **Discuss the behavior of $$ I(x) $$ at its point(s) of discontinuity.**
   - **A.** At the point(s) of discontinuity, the value(s) of the function $$ I(x) $$ is/are 5.
**Points of Discontinuity:** $$ x = 1 $$ **Function Value at Discontinuity:** 5

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