https://file.aiquickdraw.com/mathbot/file/1727118186966lj7bg3ws.jpg

The given piecewise function is:

\[
f(x) = 
\begin{cases} 
1 + x^2 & \text{if } x < 1 \\
9x - 7 & \text{if } x \geq 1
\end{cases}
\]

### Step 1: Graph the function
1. For \( x < 1 \), the function is \( f(x) = 1 + x^2 \), a parabola opening upwards with vertex at \( (0, 1) \).
2. For \( x \geq 1 \), the function is \( f(x) = 9x - 7 \), a straight line with slope 9 and y-intercept -7.

### Key Points for the Graph:
- For \( x = 1 \):
  - From the first part of the piecewise function, \( f(1^-) = 1 + 1^2 = 2 \) (as \( x \) approaches 1 from the left).
  - From the second part of the piecewise function, \( f(1^+) = 9(1) - 7 = 2 \) (as \( x \) approaches 1 from the right).
  
Both values are the same at \( x = 1 \), so the function is continuous at \( x = 1 \). Thus, the graph will have no jump or break.

### Continuity:
The function is continuous at \( x = 1 \) because:
\[
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) = 2
\]
This shows that the function is continuous for all real numbers.

Let me know if you'd like a graphical sketch of the function or more details on any step!

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Here are 5 related questions:
1. How do you determine if a piecewise function is continuous at a specific point?
2. What are the conditions for continuity of a function at a given point?
3. How do you differentiate a piecewise function at the point where the pieces meet?
4. What is the significance of left-hand and right-hand limits in continuity?
5. Can a function be continuous but not differentiable at a point?

**Tip:** When checking continuity, always compare the left-hand limit, right-hand limit, and the function's value at the point in question!

Suppose that the function f is defined for all real numbers as follows: f(x) = { 1+x^2 if x < 1, 9x-7 if x >= 1 }. Graph the function f. Then determine whether or not the function is continuous.

Explore how to graph the piecewise function f(x) = { 1+x^2 if x < 1, 9x-7 if x >= 1 } and determine its continuity. The function includes both parabolic and linear sections, and demonstrates continuity at the point x = 1.

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