Given `f(x) = 1-|4-x^2|`

show the **y-intercept**, natural **domain** and **range**.

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The function f(x) = 1 - |4 - x^2|

At the y-intercept, x = 0, 1 - |4 - x^2| = 1 - 4 = -3. The y-intercept is (0, -3)

The domain of the function is the set of real numbers R. The range is the set `[1, -oo}`

Since

`|x|=x if x>=0`

`|x|=-x if x<0`

so

`f(x)=1-(4-x^2) if (4-x^2)>=0`

`=1-4+x^2`

`=-3+x^2`

and

`f(x)=1-(-(4-x^2)) if (4-x^2)<0`

`=5-x^2`

i.e

`f(x)=-3+x^2 if x in[-2,2]`

`` `f(x)=5-x^2 if x in (-oo,-2)U(2,oo)`

Because `0 in[-2,2]` , so y intercept is

`f(0)=-3+0^2=-3`

**Thus y intercept is -3**

because f(x) is defined for all real values of x so domain of f = set of real numbers= R

Range of f = `(-oo,1]` because when `x=+-2 ,` f(x)=1,

for remaing values it will be negative.

These all above discussion you can see in graph below.

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