When looking at a rational function, Charles and Bobby have two different thoughts. Charles says that the function is defined at x = −2, x = 3, and x = 5. Bobby says that the function is undefined at those x values. Describe a situation where Charles is correct, and describe a situation where Bobby is correct. Is it possible for a situation to exist where they are both correct? Justify your reasoning.

Question

ilovecomputers1 · Answer

Y=X is an example of a case where Charles would be correct i think. But i dont get bobby

anonymous · Answer

Any rational function that has a factor of $(x-k)$ in the denominator will be undefined for $x=k$.

For examples, $\dfrac{1}{x+2}$ is undefined when $x=-2$. So any function that looks of this form will work for Charles' claim.

The opposite is true for Bobby. You can use any function that doesn't have such a factor in the denominator.

One thing to note for Charles is that a function like
$$\frac{x^2-4}{x+2}$$
will not work, even though you can remove the discontinuity at $x=-2$ by dividing:
$$\frac{x^2-4}{x+2}=\frac{(x-2)(x+2)}{x+2}=x-2$$
HOWEVER, this algebra only works for $x\not=-2$, so in fact you would have
$$\frac{x^2-4}{x+2}=\begin{cases}x-2&\text{for }x\not=-2\\text{undefined}&\text{for }x=-2\end{cases}$$