need help on functions?
f(x) = x^2
g(x)= the square root of x
the problem is that I keep finding that the 2 composite functions are equal (to x) when my textbook clearly says that 2 composite functions cannot be equal. I really need help.
Thank you soooo much!!
Favorite Answer
f(x) = x^3
g(x)= cube root of x
f(g(x)) = f(cube root of x)
= (cube root of x)^3
= x
g(f(x)) = g(x^3)
= cube root of (x^3)
= x
Therefore, f(g(x)) = g(f(x)).
In both cases the domain is all real numbers.
This proves that f and g are inverses of each other.
In the first example that you give f(g(x)) does not equal g(f(x)).
f(g(x)) = f(sqrt(x))
= (sqrt(x))^2
= x
g(f(x)) = sqrt(f(x))
= sqrt(x^2)
= |x|
Since x ≠ |x|, then f(g(x)) ≠ g(f(x))
The domain of f(g(x)) is all numbers greater than or equal to 0
(x ≥ 0).
The domain of g(f(x)) is all real numbers.
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