Question

Thanks steve for a brilliant answer! Got one last one which is gobbledegook to me.

Two functions f and g are defined by
f:x(arrow)x^2 + 3
g:x(arrow)2x + 1

find:
a) the function fg
b)solve the equation f(x) = 12g^-1(x)

Many Thanks!!!!

Answer 1

using the usual notation,

f(x) = x^2+3
g(x) = 2x+1

fg = f(x)*g(x) = (x^2+3)(2x+1) = 2x^3 + x^2 + 6x + 3

f(x) = 12g^-1(x)
g^-1(x) = (x-1)/2
so, solve x^2+3 = 12(x-1)/2
x^2+3 = 6x-6
x^2 - 6x + 9 = 0
(x-3)^2 = 0
x = 3

Answer 2

If by fg you meant f◦g (that is, f(g(x))), then

since f(x) = x^2+3,
f(g) = g^2+3 = (2x+1)^2+3 = 4x^2 + 4x + 4