CBSE Class 12 · 2023 · Relations And Functions XIILong Answer

A function

f : [- 4, 4] \to [0, 4]

is given by

f (x) = 16 - x^{2}

. Show that

f

is an onto function but not a one-one function. Further, find all possible values of

α

for which

f (α) = 7

Answer: $f$ is onto but not one-one; $α = \pm 3$

Solution

Onto: take any

y \in [0, 4]

and solve

16 - x^{2} = y

16 - x^{2} = y^{2} \Rightarrow x^{2} = 16 - y^{2} \Rightarrow x = \pm 16 - y^{2}

Since

0 \leq y \leq 4

, we have

0 \leq 16 - y^{2} \leq 16

, so

x = \pm 16 - y^{2} \in [- 4, 4]

exists in the domain. Every

y

in the codomain is attained, so

f

is onto.
Not one-one: for a value such as

y = 1

, both

x = 15

and

x = - 15

lie in

[- 4, 4]

and give

f (x) = 1

. Distinct inputs share an image, so

f

is not one-one.
Now solve

f (α) = 7

16 - α^{2} = 7 \Rightarrow 16 - α^{2} = 7 \Rightarrow α^{2} = 9

α = \pm 3

CBSE Class 12 Board · 2023 · Subjective