Find the value of
c
that makes
f
(
x
)
continuous at
x
=
2
:
f
(
x
)
=
{
x
2
+
c
x
≤
2
2
x
−
1
x
>
2

To find the value of \( c \) that makes the function \( f(x) \) continuous at \( x = 2 \), we need to ensure that the left-hand limit and the right-hand limit of \( f(x) \) as \( x \) approaches 2 are equal, and that they are equal to the value of the function at \( x = 2 \).

### Given Function

\[
f(x) = 
\begin{cases} 
x^2 + c & \text{if } x \leq 2 \\
2x - 1 & \text{if } x > 2 
\end{cases}
\]

### Steps to Solve

1. **Find the value of \( f(x) \) at \( x = 2 \) using the first piece of the function:**

\[
f(2) = 2^2 + c = 4 + c
\]

2. **Find the limit of \( f(x) \) as \( x \) approaches 2 from the left (using \( x^2 + c \)):**

\[
\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 + c) = 2^2 + c = 4 + c
\]

3. **Find the limit of \( f(x) \) as \( x \) approaches 2 from the right (using \( 2x - 1 \)):**

\[
\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x - 1) = 2 \cdot 2 - 1 = 4 - 1 = 3
\]

4. **Set the left-hand limit equal to the right-hand limit for continuity at \( x = 2 \):**

\[
4 + c = 3
\]

5. **Solve for \( c \):**

\[
c = 3 - 4 = -1
\]

### Conclusion

The value of \( c \) that makes the function \( f(x) \) continuous at \( x = 2 \) is 

\[
\boxed{-1}.
\]

Would you like more details on any part of the solution? Here are a few related questions you might consider:

1. How would the solution change if the function were discontinuous at a different point?
2. Can we apply this method to functions with more than two pieces?
3. What are the different types of discontinuities in a function?
4. How can we determine if a function is differentiable at a point?
5. What is the significance of the Intermediate Value Theorem in continuous functions?

**Tip:** To check if a function is continuous at a specific point, always ensure that the left-hand limit, right-hand limit, and the value of the function at that point are all equal.

Discover how to find the value of c that ensures continuity in the piecewise function f(x). This problem involves understanding continuity, limits, and piecewise functions, essential topics for advanced high school students.

Math Problem Statement

Find the value of c that makes f ( x ) continuous at x

2 : f ( x )

Solution

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