https://file.aiquickdraw.com/mathbot/file/173056482562928rx5emb.jpg

Let's go through the problem step-by-step.

### Given
The function \( f(x) \) is defined piecewise as:
\[
f(x) = 
\begin{cases} 
x^2, & x < 1 \\ 
a, & x = 1 \\ 
bx + 2, & x > 1 
\end{cases}
\]

### Problem
1. **Find values of \( a \) and \( b \) that make \( f(x) \) continuous at \( x = 1 \).**
2. **Once \( a \) and \( b \) are determined, plot the graph of \( f(x) \).**

### Solution

#### Part (a): Continuity at \( x = 1 \)

For \( f(x) \) to be continuous at \( x = 1 \), the left-hand limit, right-hand limit, and the function's value at \( x = 1 \) must all be equal.

1. **Left-hand limit as \( x \to 1^- \):**
   \[
   \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1^2 = 1
   \]

2. **Right-hand limit as \( x \to 1^+ \):**
   \[
   \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (bx + 2) = b(1) + 2 = b + 2
   \]

3. **Value at \( x = 1 \):**
   \[
   f(1) = a
   \]

For continuity at \( x = 1 \), these three expressions must be equal:
\[
1 = a = b + 2
\]

So we have two equations:
1. \( a = 1 \)
2. \( b + 2 = 1 \)

Solving these:
- From the first equation, \( a = 1 \).
- Substitute \( a = 1 \) into the second equation:
  \[
  b + 2 = 1 \Rightarrow b = -1
  \]

Therefore, the values that make \( f(x) \) continuous at \( x = 1 \) are:
\[
a = 1 \quad \text{and} \quad b = -1
\]

#### Part (b): Graph of \( f(x) \)

With \( a = 1 \) and \( b = -1 \), we can rewrite the function \( f(x) \) as:
\[
f(x) = 
\begin{cases} 
x^2, & x < 1 \\ 
1, & x = 1 \\ 
- x + 2, & x > 1 
\end{cases}
\]

1. For \( x < 1 \), the function is \( f(x) = x^2 \), which is a parabolic curve.
2. At \( x = 1 \), the function has the value \( f(1) = 1 \).
3. For \( x > 1 \), the function is \( f(x) = -x + 2 \), which is a linear function with a negative slope.

Would you like me to plot this graph for you, or do you have any questions on the solution so far?

---

### Follow-up Questions
1. What does it mean for a function to be continuous at a certain point?
2. How do we determine the left-hand and right-hand limits of a piecewise function?
3. Can there be a discontinuity at \( x = 1 \) if we change the values of \( a \) and \( b \)?
4. What happens to the graph if we change the values of \( a \) and \( b \) again?
5. How would you check continuity for other types of functions?

#### Tip
For a piecewise function, always examine each interval and boundary separately to check for continuity and differentiability.

Diketahui sebuah fungsi berbentuk f(x) dengan nilai yang berbeda berdasarkan interval x, dan diberikan fungsi piecewise:
\[
f(x) = 
\begin{cases} 
x^2, & x < 1 \\ 
a, & x = 1 \\ 
bx + 2, & x > 1 
\end{cases}
\]

Pertanyaan:

1. Tentukan nilai a dan b agar fungsi f(x) kontinu di x = 1.
2. Setelah menemukan nilai a dan b, gambar grafik kurva f(x).

Find the values of a and b that make the piecewise function f(x) continuous at x = 1. This problem involves determining limits, setting conditions for continuity, and analyzing piecewise functions. Suitable for advanced high school math students.

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