If \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{3 – x}\) then find f0g and its domain and range.

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Solution:

f(x) = √x

\(g(x) = \sqrt{3-x}\)

fog(x) = ?

We know,

fog = f[g(x)] = f(\(\sqrt{3-x}\)

\(= \sqrt{\sqrt{3-x}}\)

\(= (3-x)^{\frac{1}{4}}\)

Domain:

To find domain of fog,

First we have to find domain of \(g(x) = \sqrt{3-x}\)

Domain of g(x) is

3 – x ≥ 0

i.e. x ≤ 3  i.e. A = (-∞, 3].

and domain of \(f(g(x))  =  (3-x)^{\frac{1}{4}}\)

3 – x ≥ 0

i.e. x ≤ 3  i.e. B = (-∞, 3].

Here, A∩B = (-∞, 3] is domain of fog.

Range:

\(y = (3-x)^{\frac{1}{4}}\)

4√y = 3 – x

x  = 3 – 4√y

x exists for y > 0 but domain limit is (-∞, 3]

Range will be (0, 3]

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