Question

3) Find
domain and range of the
function f(x)=√100-x².

Answer 1

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Answer 2

• Domain of the function = [ - 10 , 10 ]

• Range of the function = [ 0 , 10 ]

Given :

$\displaystyle \sf The \: function \: \: f(x) = \sqrt{100 - {x}^{2} }$

To find :

The domain and range of the function

Solution :

Step 1 of 3 :

Write down the given function

Here the given function is

$\displaystyle \sf f(x) = \sqrt{100 - {x}^{2} }$

Step 2 of 3 :

Find domain of the function

$\displaystyle \sf Let \: \: y = f(x) = \sqrt{100 - {x}^{2} }$

$\displaystyle \sf{ \implies }y = \sqrt{100 - {x}^{2} }$

$\displaystyle \sf{ \implies } {y}^{2} = 100 - {x}^{2}$

Now

$\displaystyle \sf {y}^{2} \geqslant 0$

$\displaystyle \sf{ \implies }100 - {x}^{2} \geqslant 0$

$\displaystyle \sf{ \implies } {x}^{2} \leqslant 100$

$\displaystyle \sf{ \implies } - 10 \leqslant x \leqslant 10$

$\displaystyle \sf{ \implies }x \in [ - 10 , 10 ]$

Hence domain of the function = [ - 10 , 10 ]

Step 3 of 3 :

Find range of the function

$\displaystyle \sf y = \sqrt{100 - {x}^{2} }$

$\displaystyle \sf{ \implies } {y}^{2} = 100 - {x}^{2}$

$\displaystyle \sf{ \implies } {x}^{2} = 100 - {y}^{2}$

Now for a real number x ,

$\displaystyle \sf {x}^{2} \geqslant 0$

$\displaystyle \sf{ \implies }100 - {y}^{2} \geqslant 0$

$\displaystyle \sf{ \implies } {y}^{2} \leqslant 100$

But y ≥ 0

$\displaystyle \sf{ \implies } 0 \leqslant y^2 \leqslant 100$

$\displaystyle \sf{ \implies } 0 \leqslant y \leqslant 10$

$\displaystyle \sf{ \implies }y \in [ 0 , 10 ]$

Hence range of the function = [ 0 , 10 ]

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