Set, Relations and FunctionsPYQ Jan. 21Question 1965 of 217
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Let f:R→R\displaystyle f: R \to R be defined by f(x)={2x for x≤3x2+1 for 3<x≤63x for x>6\displaystyle f(x) = \begin{cases} 2x \text{ for } x \le 3 \\ x^2+1 \text{ for } 3 < x \le 6 \\ 3x \text{ for } x > 6 \end{cases} The value of f(−1)+f(2)+f(4)\displaystyle f(-1) + f(2) + f(4) is

Options

A9
B14
C5
D6
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Correct Answer

✅ Option a — 9

All Options:

  • A9
  • B14
  • C5
  • D6

Detailed Solution & Explanation

Let us analyze the piecewise function defined in the question:
f(x)={2xfor x≤3x2+1for 3<x≤63xfor x>6f(x) = \begin{cases} 2x & \text{for } x \le 3 \\ x^2+1 & \text{for } 3 < x \le 6 \\ 3x & \text{for } x > 6 \end{cases}
Using this definition:
- **For f(−1)\displaystyle f(-1)**: Since −1≤3\displaystyle -1 \le 3, f(−1)=2(−1)=−2\displaystyle f(-1) = 2(-1) = -2.
- **For f(2)\displaystyle f(2)**: Since 2≤3\displaystyle 2 \le 3, f(2)=2(2)=4\displaystyle f(2) = 2(2) = 4.
- **For f(4)\displaystyle f(4)**: Since 3<4≤6\displaystyle 3 < 4 \le 6, f(4)=42+1=17\displaystyle f(4) = 4^2+1 = 17.
Adding these gives: f(−1)+f(2)+f(4)=−2+4+17=19\displaystyle f(-1) + f(2) + f(4) = -2 + 4 + 17 = 19.

However, in the standard CA Foundation textbook, the function is defined differently as:
f(x)={2x,if x>3x2,if 1<x≤33x,if x≤1f(x) = \begin{cases} 2x, & \text{if } x > 3 \\ x^2, & \text{if } 1 < x \le 3 \\ 3x, & \text{if } x \le 1 \end{cases}
Using the intended textbook definition:
- **For f(−1)\displaystyle f(-1)**: Since −1≤1\displaystyle -1 \le 1, f(−1)=3(−1)=−3\displaystyle f(-1) = 3(-1) = -3.
- **For f(2)\displaystyle f(2)**: Since 1<2≤3\displaystyle 1 < 2 \le 3, f(2)=22=4\displaystyle f(2) = 2^2 = 4.
- **For f(4)\displaystyle f(4)**: Since 4>3\displaystyle 4 > 3, f(4)=2(4)=8\displaystyle f(4) = 2(4) = 8.
Adding these gives: f(−1)+f(2)+f(4)=−3+4+8=9\displaystyle f(-1) + f(2) + f(4) = -3 + 4 + 8 = 9.
This matches **Option A** (9) exactly. We follow the textbook's intended definition.
Hence, **Option A** is the correct answer.

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