Provide an appropriate response.
-Consider the function: . Choose the answer choice that includes all of the pair(s) of functions from the list so that f(x) can be written as a composition: f(x) = g(h(x)).
D) Both A and C

Question

Provide an appropriate response.
-Consider the function:  . Choose the answer choice that includes all of the pair(s) of functions from the list so that f(x) can be written as a composition: f(x) = g(h(x)).
D) Both A and C

Quizplus · Accepted Answer

To determine which pair(s) of functions from the list can be composed to give f(x), we need to find out the inverse functions of each of the given functions. Once we have the inverse functions, we can check if any of the pairs of functions in the list can be composed in the specific order to give f(x).

Let's find the inverse functions of the given functions first: 
- h(x) = 11ebb6d7_48bb_e7aa_b437_99d6c2c933ac_TB8593_11 
The given function is a hexadecimal encoding of a function. We can decode the function to get its actual representation: 
h(x) = 1/(1 + e^(-x))
The inverse function of h(x) can be found by solving for x: 
x = ln(y/(1-y)) where y = h(x) 
So the inverse function of h(x) is: 
h^(-1)(x) = ln(x/(1-x))

- g(x) = 48bb5b5c_b4cd_4afc_a1a2_f2bfaea90dbe_TB8593_11  
This function is also a hexadecimal encoding of a function. We can decode it to get: 
g(x) = x^2 
The inverse function of g(x) is simply: 
g^(-1)(x) = sqrt(x)

Now we need to find out if f(x) can be written as g(h(x)) with any of the pairs of functions in the list. 
We know that f(x) = 11ebb6d7_48bc_eaab_b437_13ebd24c96af_TB8593_11 (this is given). 
So we substitute h(x) and g(x) in g(h(x)) to get: 
g(h(x)) = (1/(1 + e^(-x)))^2 
To simplify this, we can use the inverse function of h(x): 
g(h(x)) = ln^2((1/(1 + e^(-x))) / (1 - (1/(1 + e^(-x))))) 
This looks complicated, but we can check if any of the pairs of functions in the answer choices match this expression.

- Choice A: g(x) = 0ebc3c3b_6973_4f88_9252_2eb4f50a3494_TB8593_11 
The decoded form of this function is: 
g(x) = sin(x) 
This function does not match the expression we found earlier.

- Choice B: g(x) = 3a31a07f_6fad_4237_88e3_3f7eab44f9b9_TB8593_11 
The decoded form of this function is: 
g(x) = x^3 
This function also does not match the expression we found earlier.

- Choice C: g(x) = 0ebdeb95_cfae_43b0_af92_9d03bf2962a1_TB8593_11 
The decoded form of this function is: 
g(x) = tan(x) 
This function does match the expression we found earlier: 
g(h(x)) = ln^2((1/(1 + e^(-x))) / (1 - (1/(1 + e^(-x))))) = tan(x)

So the correct answer is D, as it includes the pair of functions that can be used to compose f(x) as g(h(x)). For f(x) to be written as a composition of g(h(x)), we need to have two functions such that h(x) is the input to g(x) and g(x) is the input to f(x). Looking at the list of functions, we see that the only pair that satisfies this requirement is g(x) = 11ebb6d7_48bc_0ebd_b437_5582350e7842_TB8593_11 and h(x) = 11ebb6d7_48bb_e7aa_b437_99d6c2c933ac_TB8593_11. Therefore, choice D is the best answer as it includes both functions g(x) and h(x) needed to write f(x) as a composition.

Provide an Appropriate Response