Function of g is g:x→x-1 over x+2,x not equal to -2.
Find g^2
g(x) = (x-1)/(x+2)
If by g^2 you mean g(g(x)), then
g(g) = (g-1)/(g+2)
= ((x-1)/(x+2) - 1) / ((x-1)/(x+2) + 2)
= -1/(x+1)
To find g^2, we need to understand that g^2 refers to the composition of the function g with itself.
The given function g(x) = (x - 1) / (x + 2), with the condition that x is not equal to -2.
Let's start by finding g(g(x)), which is equivalent to g^2(x).
Step 1: Find g(x)
To find g(x), substitute x into the function g(x) = (x - 1) / (x + 2):
g(x) = (x - 1) / (x + 2)
Step 2: Substitute g(x) back into the function g(x) to obtain g(g(x)):
g(g(x)) = [(x - 1) / (x + 2) - 1] / [(x - 1) / (x + 2) + 2]
Step 3: Simplify g(g(x)) expression:
To simplify this expression, we need to obtain a common denominator:
g(g(x)) = [((x - 1) - (x + 2)(x - 1)) / (x + 2)] / [((x - 1) + 2(x + 2)) / (x + 2)]
g(g(x)) = [(x - 1 - (x^2 - 3x + 2)) / (x + 2)] / [(x - 1 + 2x + 4) / (x + 2)]
g(g(x)) = [(x - 1 - x^2 + 3x - 2) / (x + 2)] / [(x + 2x + 3) / (x + 2)]
g(g(x)) = [(-x^2 + 4x - 3) / (x + 2)] / [(3x + 3) / (x + 2)]
Step 4: Simplify further:
To simplify further, multiply the expression by the reciprocal of the denominator:
g(g(x)) = [-x^2 + 4x - 3] / (x + 2) * (x + 2) / (3x + 3)
g(g(x)) = (-x^2 + 4x - 3)/(3x + 3)
Therefore, g^2(x) = (-x^2 + 4x - 3)/(3x + 3).