Preview Activity 5.5.1.
Consider the rational function \(r(x) = \frac{x^2 - 1}{x^2 - 3x - 4}\text{,}\) and let \(p(x) = x^2 - 1\) (the numerator of \(r(x)\)) and \(q(x) = x^2 - 3x - 4\) (the denominator of \(r(x)\)).
(a)
(b)
(c)
Define \(r(x)\) in Desmos, and evaluate the function appropriately to find numerical values for the output of \(r\) and hence complete the following tables.
| \(x\) | \(r(x)\) |
| \(4.1\) | \(\) |
| \(4.01\) | \(\) |
| \(4.001\) | \(\) |
| \(3.9\) | \(\) |
| \(3.99\) | \(\) |
| \(3.999\) | \(\) |
| \(x\) | \(r(x)\) |
| \(1.1\) | \(\) |
| \(1.01\) | \(\) |
| \(1.001\) | \(\) |
| \(0.9\) | \(\) |
| \(0.99\) | \(\) |
| \(0.999\) | \(\) |
| \(x\) | \(r(x)\) |
| \(-1.1\) | \(\) |
| \(-1.01\) | \(\) |
| \(-1.001\) | \(\) |
| \(-0.9\) | \(\) |
| \(-0.99\) | \(\) |
| \(-0.999\) | \(\) |
(d)
Why does \(r\) behave the way it does near \(x = 4\text{?}\) Explain by describing the behavior of the numerator and denominator.
(e)
Why does \(r\) behave the way it does near \(x = 1\text{?}\) Explain by describing the behavior of the numerator and denominator.
(f)
Why does \(r\) behave the way it does near \(x = -1\text{?}\) Explain by describing the behavior of the numerator and denominator.
(g)
Plot \(r\) in Desmos. Is there anything surprising or misleading about the graph that Desmos generates?
