Find two functions f and g such that (f X g)(x) = h(x). (There are many correct answers.)
h(x) = 4 / (5x + 2)^2 =
My answers were: f(x) = 4 / 5x, and
g(x) = 5x + 2
Please explain. These answers were wrong and I do not know how to fix them!!
using your answers
g(x) = (5x+2)
f (g(x)) = 4 /[5(5x+2)]
= 4/(25x+10)
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Say g(x) = 1/(5x+2)
and f(x) = 4 x^2
then f(g(x)) = 4 [ 1^2/(5x+2)^2 ]
To find two functions f and g such that their composition (f X g)(x) equals h(x), we need to consider the inverse operations of the operations present in h(x).
Looking at h(x) = 4 / (5x + 2)^2, we see that it involves the operations of:
1. Multiplying by 5 (5x)
2. Adding 2 (5x + 2)
3. Squaring (5x + 2)^2
4. Taking the reciprocal (4 / (5x + 2)^2)
So, we need to break down these operations into two functions, f(x) and g(x).
One common approach is to split the operations between f(x) and g(x) as follows:
1. Function f(x) should handle the last operation (taking the reciprocal), which means it should provide the reciprocal of its input, i.e., f(x) = 1/x.
2. Function g(x) should handle the remaining operations (multiplying by 5, adding 2, and squaring). To achieve this, we need to perform these operations in reverse order. First, squaring can be represented as multiplying a number by itself, so g(x) should square the input value by multiplying it by itself. Next, we need to subtract 2, which is the opposite of adding 2, so g(x) should subtract 2 from the input. Finally, we need to divide by 5, which is the opposite of multiplying by 5, so g(x) should divide the input by 5.
Based on this reasoning, the correct functions are:
f(x) = 1/x
g(x) = [(x - 2) / 5]^2
Therefore, the composition of f and g, (f X g)(x), should yield h(x).
Note: It is crucial to ensure that the operations are performed in reverse order for g(x).