Set, Relations and FunctionsPYQ July 21Question 1966 of 217
All Questions

The range of the function f\displaystyle f defined by f(x)=16−x2\displaystyle f(x) = \sqrt{16-x^2} is

Options

A(−4,0)\displaystyle (-4, 0)
B(−4,4)\displaystyle (-4, 4)
C[0,4]\displaystyle [0, 4]
D(+4,4)\displaystyle (+4, 4)
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option c — [0,4]\displaystyle [0, 4]

All Options:

  • A(−4,0)\displaystyle (-4, 0)
  • B(−4,4)\displaystyle (-4, 4)
  • C[0,4]\displaystyle [0, 4]
  • D(+4,4)\displaystyle (+4, 4)

Detailed Solution & Explanation

We are given the function:
f(x)=16−x2f(x) = \sqrt{16-x^2}
Let's find the domain of the function first. For the square root to be defined, the expression inside must be non-negative:
16−x2≥0  ⟹  x2≤16  ⟹  −4≤x≤416 - x^2 \ge 0 \implies x^2 \le 16 \implies -4 \le x \le 4
Thus, the domain of f(x)\displaystyle f(x) is [−4,4]\displaystyle [-4, 4].
Now, let's determine the range. For x∈[−4,4]\displaystyle x \in [-4, 4]:
- The minimum value of 16−x2\displaystyle 16 - x^2 occurs when x2\displaystyle x^2 is maximized, i.e., x2=16\displaystyle x^2 = 16 (when x=±4\displaystyle x = \pm 4):
f(±4)=16−16=0f(\pm 4) = \sqrt{16 - 16} = 0
- The maximum value of 16−x2\displaystyle 16 - x^2 occurs when x2\displaystyle x^2 is minimized, i.e., x2=0\displaystyle x^2 = 0 (when x=0\displaystyle x = 0):
f(0)=16−0=4f(0) = \sqrt{16 - 0} = 4
Since f(x)\displaystyle f(x) is continuous on the interval [−4,4]\displaystyle [-4, 4], it takes all values in the interval [0,4]\displaystyle [0, 4].
Therefore, the range of the function is [0,4]\displaystyle [0, 4].
Hence, **Option C** is the correct answer.

More Questions from Set, Relations and Functions

If A={1,2,3,4,5}\displaystyle A = \{1, 2, 3, 4, 5\}, B={2,4}\displaystyle B = \{2, 4\} and C={1,3,5}\displaystyle C = \{1, 3, 5\} then (A−C)×B\displaystyle (A - C) \times B is:

If f(x)=x2−5\displaystyle f(x)=x^2-5, evaluate f(3)\displaystyle f(3), f(−4)\displaystyle f(-4), f(5)\displaystyle f(5), and f(1)\displaystyle f(1).

Out of a group of 20\displaystyle 20 teachers in a school, 10\displaystyle 10 teach Mathematics, 9\displaystyle 9 teach Physics and 7\displaystyle 7 teach Chemistry. 4\displaystyle 4 teach Mathematics and Physics but none teach both Mathematics and Chemistry. How many teach Chemistry and Physics; how many teach only Physics?

If A={1,2,3,4,5,7,8,9}\displaystyle A = \{1, 2, 3, 4, 5, 7, 8, 9\} and B={2,4,6,7,9}\displaystyle B = \{2, 4, 6, 7, 9\} then how many proper subset of A∩B\displaystyle A \cap B can be created?

The number of subsets of the set {0,1,2,3}\displaystyle \{0, 1, 2, 3\} is:

If f(x)=x2+x−1\displaystyle f(x) = x^2+x-1 and 4f(x)=f(2x)\displaystyle 4f(x) = f(2x), then find the value of 'x'.

Ready to Master Set, Relations and Functions?

Practice all 217 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free