Preview Activity 1.6.1.
(a)
Let \(r(t) = p(q(t))\text{.}\) Determine a formula for \(r\) that depends only on \(t\) and not on \(p\) or \(q\text{.}\)
(b)
Review the introductory example with \(f(x) = x^2 - 1\) and \(g(t) = 3t - 4\text{,}\) which involved functions similar to \(p\) and \(q\) in part (a). What is the biggest difference between your work in (a) above and in the introductory example?
(c)
Let \(t = s(z) = \frac{1}{z+4}\) and recall that \(x = q(t) = t^2 - 1\text{.}\) Determine a formula for \(x = q(s(z))\) that depends only on \(z\text{.}\)
(d)
Suppose that \(h(t) = \sqrt{2t^2 + 5}\text{.}\) Determine formulas for two related functions, \(y = f(x)\) and \(x = g(t)\text{,}\) so that \(h(t) = f(g(t))\text{.}\)
