f(x)= square root of x and g(x)=^3square root 1 - x. Show the domain and fully simplify the expression.

a) f x g

b) g x f

c) f x f

For g(x) the equation reads square root of 1-x raised to the third power.

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f(x) = sqrtx = x^(1/2)

g(x) = 3^sqrt(1-x).

To determine the domain and the functions: f (g(x), and g(f(x) and f((x).

Solution:

domain:

The domain sqrt of x = x^(1/2) is x > = 0. The domain of sqrt(x-1) is x-1 >= 0. Or x > = 1. Both together the domain is x > 1.

f*g =f(g(x)

f*g = sqrt { g(x)}

f*g = sqrt { 3^sqrt(x-1)

f*g = 3^ ((1/2)sqrt(x-1)), by index law sqrta = a^(1/2).

f*g = 3 ^[0.5sqrt(x-1)]

g*f = g(f(x))

g*f =3^sqrt (f(x))

g*f = 3^(sqrt(sqrtx))

g*f = 3^(sqrt(x^(1/2)))

g*f= 3 ^(x ^((1/2)*(1/2)))

g*f = 3^(x^0.25)

f*f = f((f(x))

f*f = sqrt { sqrtx}

f*f= sqrt(x^(1/2)

f*f= x^((1/2)(1/2)) = x^0.25.

f(x) = sqrt(x)

g(x) = [sqrt(1-x)]^3 = (1-x)*sqrt(1-x)

1) f x g = (sqrtx)*(1-x)*sqrt(1-x)

= (1-x)sqrt(x(1-x))

= (1-x)*sqrt(x-x^2)

2) g X f = f X g = (1-x)*sqrt(x-x^2)

3)f *f = sqrtx * sqrtx = x

==> f X f = x

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