Question

Let f(x)=x2−7x+12 and g(x)=x2−9.

What is (fg)(x)?
x−4/x−3 where ​x≠−3,4​
x+3/x−4 where x≠3,4
x−4/x+3 where x≠−3,3
x+4/x−3 where ​x≠−3,3

Answer 1

So, we need to find the zeroes of both f and g to write them in linear-factor-form. Please understand x_(1/2) as an x with 1/2 in the index, not one half

f(x)=0
x^2-7x+12=0
x_(1/2) = 3.5 +- sqrt( 3.5^2 -12)

x_(1/2) = 3.5 +- sqrt( 12.25 -12)

x_(1/2) = 3.5 +- sqrt( 0.25)

x_(1/2) = 3.5 +- 0.5

Hence x_1 = 4 and x_2 = 3

Similarly (or by using the third binomial formula, we get the two zeroes

x_1 = -3 and x_2 = 3 for g.

That means that we can write f as
f(x) = (x-4)(x-3)
And
g(x) = (x-(-3))(x-3)=(x+3)(x-3)

You can factorize these if you need some convincing ;)

That means if we divide f by g, we can of course not use the zeros of g (because then we would divide by zero) That means x cant be neither 3 nor -3.

Furthermore, we can can cancel out the linear factor (x-3) and we get (x-4)/(x+3)