Find F(x) and G(x) such that h(x)=(f*g)(x)

Question

Find F(x) and G(x) such that h(x)=(f*g)(x)

h(x)=x^2-5/x^2+5

Answer 1

(f×g)(x), or (fg)(x), just means you're multiplying two functions together. So if I had f(x) = x and g(x) = x², then (fg)(x) = x(x²) = x³. As another example, if f(x) = (x + 19) and g(x) = (x² + 2), then (fg)(x) = (x + 19)(x² + 2) = x³ + 19x² + 2x + 38.

 

That means you're looking for two functions that, when multiplied together, equal (x² - 5) / (x² + 5).

 

Here's one option:

 

f(x) = x² - 5

g(x) = 1 / (x² + 5)

 

Then (fg)(x) = (x² - 5)(1 / (x² + 5)) = (x² - 5) / (x² + 5) = h(x)

 

 

Hope this helps!