Question

If \( f(x)= \begin{cases x^2+k, & x\leq 1 \\ 2x+1, & x>1 \end{cases} \) is continuous at \( x=1 \), then the value of \( k \) is:}

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For piecewise functions, continuity at the "break point" means the function's value, left-hand limit, and right-hand limit at that point must all be equal. This usually boils down to setting the two function expressions equal at the break point.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks to find the value of a constant \( k \) that makes a piecewise function continuous at a specific point \( x=1 \).

Step 2: Key Formula or Approach:
For a function \( f(x) \) to be continuous at a point \( x=a \), three conditions must be met:
1. \( f(a) \) must be defined.
2. \( \lim_{x \to a^-} f(x) \) (left-hand limit) must exist.
3. \( \lim_{x \to a^+} f(x) \) (right-hand limit) must exist.
4. All three must be equal: \( f(a) = \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \).

Step 3: Detailed Explanation:
Given function: \( f(x)= \begin{cases} x^2+k, & x\leq 1 2x+1, & x>1 \end{cases} \)
We need to ensure continuity at \( x=1 \).
1. Value of the function at \( x=1 \):
When \( x=1 \), we use the first part of the function definition ($x \leq 1$):
\( f(1) = (1)^2 + k = 1 + k \).
2. Left-hand limit at \( x=1 \):
\( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + k) \)
As \( x \) approaches 1 from the left (values less than 1), we use the first part of the function:
\( \lim_{x \to 1^-} (x^2 + k) = (1)^2 + k = 1 + k \).
3. Right-hand limit at \( x=1 \):
\( \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x + 1) \)
As \( x \) approaches 1 from the right (values greater than 1), we use the second part of the function:
\( \lim_{x \to 1^+} (2x + 1) = 2(1) + 1 = 2 + 1 = 3 \).
For the function to be continuous at \( x=1 \), these three values must be equal:
\( f(1) = \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) \)
So, \( 1 + k = 3 \).
Solve for \( k \):
\( k = 3 - 1 \)
\( k = 2 \).

Step 4: Final Answer:
The value of \( k \) is 2.