Write the equation of the function g(x) if g(x) = f(x – 2) +4 and f(x) = x^3 + 2.
f(x-2) = (x-2)^3 +2 =x^3-6x^2+12x-8+2
= x^3 - 6x^2 + 12 x + 6
g(x) = that + 4
= x^3 - 6x^2 + 12 x + 10
To find the equation of the function g(x), you need to substitute the expression for f(x) into the equation for g(x).
First, let's find f(x - 2), since that is a part of the equation for g(x). To do this, replace every instance of x in f(x) with (x - 2):
f(x - 2) = (x - 2)^3 + 2
Expanding this expression, you get:
f(x - 2) = (x - 2)(x - 2)(x - 2) + 2
= (x - 2)(x^2 - 4x + 4) + 2
= x(x^2 - 4x + 4) - 2(4x - 8) + 2
= x^3 - 4x^2 + 4x - 8x + 16 + 2
= x^3 - 4x^2 - 4x + 18
Now, substitute f(x - 2) into the equation for g(x):
g(x) = f(x - 2) + 4
= x^3 - 4x^2 - 4x + 18 + 4
= x^3 - 4x^2 - 4x + 22
Therefore, the equation of the function g(x) is g(x) = x^3 - 4x^2 - 4x + 22.