Under what circumstances is fog = gof always true? When is this statement true for f(x) = x2 + x – 2 and g(x) = x + 4? (4 marks)
X^2***
It is true when f(x) and g(x) are inverses of each other, that is,
f(g(x)) = g(f(x)) = x
for f(x) = x^2 + x – 2 and g(x) = x + 4, since x+4 is clearly not the inverse of f(x) fog ≠ gof
The statement "fog = gof" is true when the compositions of the functions f and g are equal for all values in their respective domains. In other words, it means that for every input x in the domain of f and g, the output of f(g(x)) is the same as the output of g(f(x)).
To determine whether fog = gof for the given functions f(x) = x^2 + x - 2 and g(x) = x + 4, we need to calculate the compositions and check if they are equal.
1. Composition f(g(x)):
We substitute g(x) into f(x):
f(g(x)) = f(x + 4) = (x + 4)^2 + (x + 4) - 2 = x^2 + 9x + 14
2. Composition g(f(x)):
We substitute f(x) into g(x):
g(f(x)) = g(x^2 + x - 2) = (x^2 + x - 2) + 4 = x^2 + x + 2
Now, let's compare the two compositions:
f(g(x)) = x^2 + 9x + 14
g(f(x)) = x^2 + x + 2
Since f(g(x)) and g(f(x)) are different, we can conclude that fog is not equal to gof for the functions f(x) = x^2 + x - 2 and g(x) = x + 4.
Therefore, under these circumstances, fog = gof is not true.