For the given functions f and g find each of the following c

For the given functions f and g, find each of the following composite functions. f(x) = x^2 + 2 g(x) = squareroot x - 2

Solution

f(x) = x² + 2

g(x) = (x - 2)

a) f o g = f(g(x))

==> f((x - 2))

==> ((x - 2) )² + 2

==> x -2 + 2

==> x

Hence f o g = x

b) g o f = g(f(x))

==> g(x² + 2)

==> ( (x² + 2) - 2)

==> x²

==> x

Hence g o f = x

c) f o f

==> f(f(x))

==> f(x² + 2)

==> (x² + 2)² + 2

==> x⁴ + 4x² + 4 + 2

==> x⁴ + 4x² + 6

Hence f o f = x⁴ + 4x² + 6

d) g o g = g(g(x))

==> g((x - 2) )

==> ( ((x - 2)) - 2)

==> ((x - 2) - 2)

Hence g o g = ((x - 2) - 2)