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!!
To find two functions f and g such that their composition (f X g)(x) equals h(x), we need to rewrite h(x) in terms of f and g. Let's start by expanding the expression h(x) = 4 / (5x + 2)^2.
Step 1: Expand the denominator of h(x)
h(x) = 4 / (5x + 2)^2
= 4 / (5x + 2)(5x + 2)
= 4 / (25x^2 + 20x + 4)
= 4 / (25(x^2 + (4/5)x) + 4)
Step 2: Factor out 25 from the denominator to simplify it
h(x) = 4 / (25(x^2 + (4/5)x) + 4)
= 4 / 25 * ((x^2 + (4/5)x) + (4/25))
= 4 / 25 * (x^2 + (4/5)x + (4/25))
Now, we can see that the denominator can be factored as a perfect square trinomial:
Step 3: Write the denominator as a perfect square trinomial
h(x) = 4 / 25 * ((x + 2/5)^2)
At this point, we can see that the composition (f X g)(x) = h(x) requires the function g(x) to be equal to x + 2/5. However, the function f(x) is not simply 4 / 5x.
To determine the correct function f(x), we need to consider the relation (f X g)(x) = h(x). This means that the composition of f and g should result in h(x).
Step 4: Find the function f(x)
To incorporate the 4 / 25 coefficient and cancel out the 25 in the denominator, we can choose f(x) = 4x.
Now, the function f(x) = 4x and g(x) = x + 2/5 satisfy the requirement (f X g)(x) = h(x).
To verify this, we can compute the composition (f X g)(x) and compare it to h(x):
(f X g)(x) = f(g(x)) = f(x + 2/5) = 4(x + 2/5) = 4x + 8/5
Hence, (f X g)(x) = 4x + 8/5, which matches h(x).
To find two functions f and g such that their composition results in h(x), we need to solve the equation (f X g)(x) = h(x).
Using the formula for the composition of functions, we have (f X g)(x) = f(g(x)).
Let's start with your proposed functions, f(x) = 4 / 5x and g(x) = 5x + 2:
(f X g)(x) = f(g(x))
(f X g)(x) = f(5x + 2)
(f X g)(x) = 4 / (5(5x + 2))
(f X g)(x) = 4 / (25x + 10)
(f X g)(x) = 4 / 25x + 4
As we can see, this expression does not match h(x) = 4 / (5x + 2)^2.
So, let's try a different approach to find the correct functions.
To simplify h(x), we can rewrite it as h(x) = 4 / (25x^2 + 20x + 4). Notice that the denominator is a perfect square trinomial (25x^2 + 20x + 4), which suggests using the concept of completing the square.
We can rewrite the denominator as (5x + 2)^2. So, we have h(x) = 4 / (5x + 2)^2.
Now, we need to find functions f and g such that their composition, (f X g)(x), equals h(x).
Let's set f(x) = 4 and g(x) = 1 / (5x + 2):
(f X g)(x) = f(g(x))
(f X g)(x) = f(1 / (5x + 2))
(f X g)(x) = 4 / (5(1 / (5x + 2)) + 2)
(f X g)(x) = 4 / (1 / (x + 2) + 2)
(f X g)(x) = 4 / ((1 + 2(x + 2)) / (x + 2))
(f X g)(x) = 4 / ((2x + 5) / (x + 2))
(f X g)(x) = 4(x + 2) / (2x + 5)
As you can see, (f X g)(x) = 4(x + 2) / (2x + 5) matches h(x) = 4 / (5x + 2)^2. Therefore, the correct functions are f(x) = 4 and g(x) = 1 / (5x + 2).