How do you determine the three functions f, g, and h such that
#(f o g o h)(x) = sin^2(x+1)#?

Question

Anna Allen · Accepted Answer

undefined To determine the functions $ f $, $ g $, and $ h $ such that $(f \circ g \circ h)(x) = \sin^2(x+1)$, you need to consider the composition of functions and how they relate to the given expression.

Given that $ \sin^2(x+1) $ is a composite function, we can break it down into simpler components.

1. The innermost function $ h(x) $ should produce the argument $ x+1 $ for the sine function.
2. The middle function $ g(x) $ should produce $ \sin(x+1) $ from the output of $ h(x) $.
3. The outermost function $ f(x) $ should square the output of $ g(x) $, resulting in $ \sin^2(x+1) $.

Therefore, the functions $ f(x) $, $ g(x) $, and $ h(x) $ can be determined as follows:

1. Let $ h(x) = x + 1 $.
2. Let $ g(x) = \sin(h(x)) = \sin(x + 1) $.
3. Let $ f(x) = g(x)^2 = (\sin(x + 1))^2 $.

So, the functions are:

$$ h(x) = x + 1 $$
$$ g(x) = \sin(x + 1) $$
$$ f(x) = (\sin(x + 1))^2 $$

Liam Brennan · Answer

Observe what is being done to #x# to find that
#h(x) = x+1#
#g(x) = sin(x)#
#f(x) = x^2# Let's look at what we're doing to #x# in #sin^2(x+1)#, and what order it is being done in. Then, remembering that when applying #f@g@h# we start with #h# and then work to the left, the functions must be #h(x) = x+1# #g(x) = sin(x)# #f(x) = x^2#

How do you determine the three functions f, g, and h such that #(f o g o h)(x) = sin^2(x+1)#?