Find (fog)(x)and (gof)(x)if f(x)=x-x^2 and g(x)= 2x+3.
For f(of)g(x), take (2x+3) - (2x+3)^2
(insert g(x) where the "x" is in f(x) )
For g(of)f(x), take 2[x-x^2] +3
To find (fog)(x) (also known as f composed with g) and (gof)(x) (also known as g composed with f), we need to substitute the expression for g(x) into f(x) and substitute the expression for f(x) into g(x).
Start with (fog)(x):
1. Replace every instance of "x" in f(x) with the expression for g(x):
fog(x) = f(g(x))
= f(2x + 3)
2. Substitute 2x + 3 into f(x):
fog(x) = (2x + 3) - (2x + 3)^2
= (2x + 3) - (4x^2 + 12x + 9)
= 2x + 3 - 4x^2 - 12x - 9
= -4x^2 - 10x - 6
Now, let's find (gof)(x):
1. Replace every instance of "x" in g(x) with the expression for f(x):
gof(x) = g(f(x))
= g(x - x^2)
2. Substitute x - x^2 into g(x):
gof(x) = 2(x - x^2) + 3
= 2x - 2x^2 + 3
Therefore, (fog)(x) = -4x^2 - 10x - 6 and (gof)(x) = 2x - 2x^2 + 3.