Determine the domain and the range of the function $f (x) = 5 - x^{2}$ .

medium 2 marks Class 10 · Mathematics · Real Numbers

Answer

Domain:

[- s q r t 5, s q r t 5]

; Range:

[0, s q r t 5]

The correct answer is Domain: $[- s q r t 5, s q r t 5]$ ; Range: $[0, s q r t 5]$ — Hence the range is $[0, s q r t 5]$ .

Step-by-step solution

Require the radicand to be non‑negative: $5 - x^{2} g e 0$ .
Rearrange: $x^{2} l e 5$ .
Take square roots: $- s q r t 5 l e x l es q r t 5$ .
Thus the domain is $[- s q r t 5, s q r t 5]$ .
For any $x$ in this interval $f (x) = 5 - x^{2}$ ranges from $0$ (when $x = p m s q r t 5$ ) to $s q r t 5$ (when $x = 0$ ).
Hence the range is $[0, s q r t 5]$ .

Common mistake: Forgetting negative

x

values or mis‑identifying the maximum value

Concept tested: Domain and range of radical function