find the functions f&g so that fog = H
H(x)=squre root x^2+8
please show work
If you mean H(x) = sqrt(x^2 + 8), the f and g functions could be
g(x) = x^2 +8
and
f(x) = sqrt(x)
But there are other possibilities. For example
g(x) = (x^2 + 8)^2 and
f(x) = x^1/4
To find the functions f and g such that fog = H, we need to use function composition.
Let's start by writing out the functions f and g in terms of their inputs. Let f(x) = a(x) and g(x) = b(x).
According to the equation fog = H, we can say that f(g(x)) = H(x). Substituting f(x) and g(x) with their corresponding expressions, we get a(b(x)) = √(x^2 + 8).
Now, let's break down the equation further. Since a(b(x)) = √(x^2 + 8), we can write a(b(x))^2 = x^2 + 8.
Next, we can consider the inner function g(x) = b(x). Squaring both sides of the equation, we have b(x)^2 = x^2 + 8.
Now, let's solve for b(x). Taking the square root of both sides, we find b(x) = √(x^2 + 8).
Moving on to the outer function f(x) = a(x), we know that f(g(x)) = a(b(x)). Plugging in the value of b(x), we have a(√(x^2 + 8)).
Therefore, the functions f and g that satisfy fog = H are:
f(x) = a(x)
g(x) = √(x^2 + 8)
Note that the specific form of function a(x) is not determined in this equation. It can be any function that takes the input x.