Find f(x) and g(x) such that h(x)=(fog)(x)
h(x)=x^2-9 / x^2+9
one way would be
g(x) = x^2+9
f(x) = (x-18)/x
so, (fog)(x) = f(g) = (g-18)/g = (x^2-9)/(x^2+9)
There are other ways, too, of course.
To find f(x) and g(x) such that h(x) = (fog)(x), we need to decompose the function h(x) into the composition of two separate functions, f(x) and g(x).
Let's first rewrite h(x) as:
h(x) = (x^2-9)/(x^2+9)
To decompose h(x) into two functions, let's assume g(x) = x^2+9, since the denominator of h(x) is x^2+9.
Now, we can find f(x) by substituting g(x) into h(x) and solving for f(x):
h(x) = (fog)(x)
(x^2-9)/(x^2+9) = f(x^2+9)
To isolate f(x), we can multiply both sides of the equation by (x^2+9):
(x^2-9) = f(x^2+9) * (x^2+9)
Next, we can simplify and solve for f(x):
(x^2-9) = f(x^2+9) * (x^2+9)
(x^2-9)/(x^2+9) = f(x^2+9)
Now, we have f(x) = (x^2-9)/(x^2+9). Therefore, g(x) = x^2+9 and f(x) = (x^2-9)/(x^2+9) are the required functions to form the composition h(x) = (fog)(x).