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Is the graph of this function continuous?

To determine if the given piecewise function is continuous, we need to check for continuity at the points where the function changes its definition, specifically at \( x = 1 \). The function is defined as:

\[
f(x) =
\begin{cases}
x^2 + 1 & \text{if } x < 1 \\
2 & \text{if } x = 1 \\
7 - 5x & \text{if } 1 < x
\end{cases}
\]

### Step 1: Check Continuity at \( x = 1 \)

A function is continuous at \( x = 1 \) if the following conditions are met:
1. \( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) \) (The left-hand limit equals the right-hand limit).
2. The value of the function at \( x = 1 \) exists and equals the limit from both sides.

#### Left-hand limit (\( x \to 1^- \)):

For \( x < 1 \), the function is \( f(x) = x^2 + 1 \). Taking the limit as \( x \) approaches 1 from the left:
\[
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + 1) = 1^2 + 1 = 2
\]

#### Right-hand limit (\( x \to 1^+ \)):

For \( x > 1 \), the function is \( f(x) = 7 - 5x \). Taking the limit as \( x \) approaches 1 from the right:
\[
\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (7 - 5x) = 7 - 5(1) = 2
\]

#### Function value at \( x = 1 \):

The function value at \( x = 1 \) is directly given as:
\[
f(1) = 2
\]

### Conclusion:
- \( \lim_{x \to 1^-} f(x) = 2 \)
- \( \lim_{x \to 1^+} f(x) = 2 \)
- \( f(1) = 2 \)

Since the left-hand limit, right-hand limit, and the function value all equal 2, the function is continuous at \( x = 1 \).

### Final Answer:
Yes, the function is continuous.

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Would you like more details or have any questions? Here are some related questions for further exploration:

1. What would happen if \( f(1) \neq \lim_{x \to 1} f(x) \)?
2. How would the continuity of this function change if the middle piece were different?
3. What is the significance of continuity in piecewise functions?
4. How would you check for continuity at multiple points in a function?
5. What role do limits play in determining continuity?

**Tip:** To ensure a function is continuous at a specific point, always check both the left-hand and right-hand limits, as well as the function's value at that point.

Determine the continuity of a piecewise function at x = 1 by calculating the left-hand and right-hand limits and comparing them to the function's value at that point. This solution involves concepts of limits, continuity, and piecewise functions. Suitable for Grades 10-12.

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