I'm trying to solve a function problem. It states:
Determine the values of $a$, $b$, and $d$ so that the rational function $$f(x) = \frac{(x+a)(x-1)(x-b)}{(x-c)(x+d)(x-3)}$$
correctly models this graph:
I've been looking at this for a while and I just can't figure out how I'm supposed to approach it. I think it has something to do with limits and I know they're all going to be integer numbers. Any ideas beyond randomly plugging numbers into Desmos until I get the right graph?
Best Answer
from the expression, we see that whenever the numerator is zero, the graph intersects $x$ axis, and the numerator becomes zero when either of the factors is zero hence possible roots: $x=-a, x=1$ and $x=b $. Also when the denominator is zero, the graph should tend to infinity if the numerator is non-zero at the same time. ie: $x= c, x=-d$ and $x=3$.
From graph, roots are $x=-2,x=1$. Hence hints for $a$ and $b$.
Also as the discontinuity is at $x=3$, the function is not defined there hence the denominator is zero but the numerator is also but because nothing can be divided by zero, it is not defined at that single point. So you have to think of ways in which we can choose $c$ and $d$ such that at one of the function is not defined and at the other it simply reaches infinity.
You can take it from here, I guess.