Given:
f(x)=2x^2-x+1
g(x)=2 sin x
h(x)=3^x
Determine:
a) f(g(x))
b) (h^-1 o f)(x)
c) g(f(h(x))))
Please show steps, thank you very much.
To determine the compositions of functions f(g(x)), (h^-1 o f)(x), and g(f(h(x))), we need to substitute the expressions g(x), h(x), and f(x) into the corresponding functions.
a) f(g(x)): We substitute g(x) = 2 sin x into f(x).
f(g(x)) = f(2 sin x)
= 2(2 sin x)^2 - (2 sin x) + 1
Simplifying further:
= 2(4 sin^2 x) - 2 sin x + 1
= 8 sin^2 x - 2 sin x + 1
Therefore, f(g(x)) = 8 sin^2 x - 2 sin x + 1.
b) (h^-1 o f)(x): To solve this, we need to find the inverse of h(x) first.
Finding the inverse of h(x):
We can rewrite h(x) as y = 3^x.
Swap the x and y variables: x = 3^y.
Solve for y: Take the logarithm of both sides.
log(x) = log(3^y)
log(x) = y log(3)
y = log(x) / log(3)
Therefore, h^-1 (x) = log(x) / log(3).
Now, we substitute f(x) = 2x^2 - x + 1 into h^-1 (x).
(h^-1 o f)(x) = h^-1 (f(x))
= log(f(x)) / log(3)
= log(2x^2 - x + 1) / log(3)
Therefore, (h^-1 o f)(x) = log(2x^2 - x + 1) / log(3).
c) g(f(h(x))): We substitute h(x) = 3^x into f(x), and then substitute the result into g(x).
g(f(h(x))) = g(f(3^x))
= g(2(3^x)^2 - (3^x) + 1)
Simplifying further:
= g(2(9^x) - 3^x + 1)
= g(18^x - 3^x + 1)
Therefore, g(f(h(x))) = g(18^x - 3^x + 1).
These are the steps to determine the compositions of the given functions.