Today's activity introduces mathematical "zooming".

1. Suppose that \(f(x) = \dfrac{2 x^2-x-6}{x-2}, x\neq2\).

- If \(6 < f(x) < 8\), what domain makes \(f(x)\) go off the "sides" of the calculator window?

What is the biggest window you can make? Hint: Stay close to \(x = 2\).

\( 1.5 \lt x \lt 2.5 \) - If \(6.9 < f(x) < 7.1\), what domain makes \(f(x)\) go off the "sides" of the calculator window? What is the biggest window you can make?

\( 1.95 \lt x \lt 2.05 \) - If \(6.99 < f(x) < 7.01\), what domain makes \(f(x)\) go off the "sides" of the calculator window? What is the biggest window you can make?

\( 1.995 \lt x \lt 2.005 \) - Without guess and checking, how could you find the "biggest" domains above?

Set 2x + 3 equal to lower/upper bound and solve for x

2. Suppose that \(g(x) = \dfrac{x}{x-2}, x \neq 2\).

- If \(-2 < g(x) < 0\), what domain makes \(g(x)\) go off the "sides" of the calculator window? What is the biggest window you can make? Hint: Stay close to \(x=1\).

\( 0 \lt x \lt \frac{4}{3} \)

\( \frac{2}{3} \lt x \lt \frac{4}{3} \) - If \(-1.1 < g(x) < -0.9\), what domain makes \(g(x)\) go off the "sides" of the calculator window? What is the biggest window you can make?

\( \frac{18}{19} \lt x \lt \frac{22}{21} \)

\( \frac{20}{21} \lt x \lt \frac{22}{21} \) - If \(-1.01 < g(x) < -0.99\), what domain makes \(g(x)\) go off the "sides" of the calculator window? What is the biggest window you can make?

\( \frac{198}{199} \lt x \lt \frac{202}{201} \)

\( \frac{200}{201} \lt x \lt \frac{202}{201} \) - For parts a-c, what are the biggest windows that will work that are
**symmetrical**to \(x=1\)? For example, \([0,3]\) is not symmetrical to \(x=1\), but \([0,2]\) is.