1.

Find the domain of each of the following real valued functions of realvariable:`f(x)=sqrt(x-2)`(ii) `f(x)=1/(sqrt(x^2-1))`(iii) `f(x)=sqrt(9-x^2)`(iv) `f(x)=sqrt((x-2)/(3-x))`

Answer» (i) `f(x) = sqrt(x-2)`
As, anything under square root can not be negative,
`:. x - 2 ge 0 => x ge 2`
So, domain of `f(x)` is `x in [2,oo]`.

(ii) `f(x) = 1/sqrt(x^2-1)`
As, anything under square root can not be negative and denominator can not be zero,
`:. x^2-1 gt 0=> (x-1)(x+1) gt 0`
So, domain of `f(x)` is `x in (-oo,-1) uu (1,oo)`.

(iii) `f(x) = sqrt(9-x^2)`
As, anything under square root can not be negative,
`:. 9-x^2 ge 0 =>(3-x)(3+x) ge 0 =>(x-3)(x+3) le 0`
So, domain of `f(x)` is `x in [-3,3]`.

(iv) `f(x) = sqrt((x-2)/(3-x))`
As, anything under square root can not be negative and denominator can not be zero,
`:. x-2 ge 0 and 3-x gt 0`
`=> x ge 2 and x lt 3`
So, domain of `f(x)` is `x in [2,3)`.


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