Given function f(x)=4x3-5, g(x)=2x and h(x)=x2. Find composition of h◦(g◦f)
f(x)=4x³
g(x)=2x
h(x)=x²
h◦(g◦f)
=h◦(g(x²))
=h◦(2(x²))
=h◦(2x²)
=4(2x²)³-5
=4(2²x²)³;-5
=4(2^6 x^6)-5
=256x^6-5
h◦(g◦f)
= h(g(f(x) ) )
= h(g(4x^3 - 5) ) )
= h(2(4x^3 - 5))
= ( 2(4x^3 - 5) )^2
manipulate this if needed.
testing with some value, say x = 2
according to my result:
(2(32-5))^2
= 2916
long way:
f(2) = 27
g(27) = 54
h(54) = 2916
Oops, I didn't do what the question asked. Go with Reiny's solution!
To find the composition of functions h◦(g◦f), we need to first evaluate the innermost composition, which is g◦f. This means we substitute the function f(x) into g(x) and simplify. Then, we substitute the resulting function into h(x) and simplify further.
Let's start by finding the composition g◦f:
g(x) = 2x
f(x) = 4x^3 - 5
Substituting f(x) into g(x), we get:
g◦f(x) = g(f(x)) = 2(f(x)) = 2(4x^3 - 5)
Simplifying further:
g◦f(x) = 2(4x^3 - 5) = 8x^3 - 10
Now, let's find the composition h◦(g◦f):
h(x) = x^2
Substituting g◦f(x) into h(x), we get:
h◦(g◦f)(x) = h(g◦f(x)) = h(8x^3 - 10)
Simplifying further:
h◦(g◦f)(x) = (8x^3 - 10)^2 = (64x^6 - 160x^3 + 100)
Therefore, the composition h◦(g◦f) is equal to 64x^6 - 160x^3 + 100.