Given the graph of the rational function:
As shown there are two asymptotes
The lines of the asymptotes are:
x = -3, x = 2
So. the factors of the denominator are:
[tex](x+3),(x-2)[/tex]
Note: (x - 2) will be repeated
And the function has an x-intercept at the point x = 1
So, the factor of the numerator will be (x - 1)
So, the function will have the following form:
[tex]f(x)=\frac{a(x-1)^2}{(x+3)(x-2)(x-2)}[/tex]
we will find the value of (a) using the point of the y-intercept
As shown the y-intercept = (0, -1/2)
So, substitute with x = 0, f = -1/2
[tex]\begin{gathered} -\frac{1}{2}=\frac{a\cdot(-1)^2}{3\cdot(-2)\cdot(-2)} \\ a=-\frac{1}{2}\cdot3\cdot(-2)\cdot(-2)=-6 \end{gathered}[/tex]
So, the answer will be:
The numerator is: -6 (x - 1)²
The denominator is: ( x + 3) ( x - 2 ) (x - 2)
