Find two functions f and g such that (f X g)(x) = h(x). ( There are many correct answers.
My answers were f(x) = 4/5x and g(x) = 5x + 2 but this was wrong.
I do not know how to do this.
To find two functions, f and g, such that their composition (f X g)(x) equals a given function h(x), we need to work backwards. We will start with h(x) and find f(x) and g(x) that satisfy the equation.
Given that (f X g)(x) = h(x), we can break down the composition into two steps:
Step 1: Determine the function g(x)
To find g(x), we need to analyze the composition equation (f X g)(x) = h(x) and isolate g(x) on one side. Let's rearrange the equation:
(f X g)(x) = h(x)
f(g(x)) = h(x)
Now, we can see that g(x) is found inside the function f(x). To isolate g(x), we need to "undo" the effect of f(x) on g(x) by taking the inverse function of f(x) or finding an operation that undoes the effect of f(x) on g(x).
Step 2: Determine the function f(x)
Once g(x) is isolated, we can easily determine f(x) by substituting the expression for g(x) back into the original composition equation:
f(g(x)) = h(x)
f(expression for g(x)) = h(x)
Now, let's work through an example:
Suppose, h(x) = 3x^2 + 1. We need to find f(x) and g(x) such that (f X g)(x) = h(x).
Step 1: Determine the function g(x):
To isolate g(x), we need to find the inverse of f(x) or an operation that undoes the effect of f(x) on g(x). In this case, if f(x) = 4/5x and g(x) = 5x + 2, we can evaluate (f X g)(x) as follows:
(f X g)(x) = f(g(x)) = f(5x + 2)
= (4/5)(5x + 2) = 4x + 8/5
However, h(x) = 3x^2 + 1, so our choice of g(x) = 5x + 2 does not produce the correct h(x) result.
To find a suitable g(x), we should consider the composition equation and think about a function that can transform the input differently.
Step 2: Determine the function f(x):
Now, substitute the expression for g(x) back into the original composition equation:
f(g(x)) = h(x)
f(5x + 2) = 3x^2 + 1
From here, we can solve for f(x) by isolating it. Considering the given values, we can find f(x) as follows:
f(5x + 2) = 3x^2 + 1
Since we can choose many functions f(x) that will satisfy this equation, let's select a simple one, such as f(x) = x:
f(x) = x
Now, we have g(x) = 5x + 2 and f(x) = x, which satisfy the composition equation.
Thus, (f X g)(x) = h(x), where f(x) = x, g(x) = 5x + 2, and h(x) = 3x^2 + 1.