Chapter 1: Problem 2
In Exercises \(1-4,\) find the coordinate increments from \(A\) to \(B\) $$A(-3,2), \quad B(-1,-2)$$
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Let \(y_{1}=x^{2}\) and \(y_{2}=2^{x}\) . (a) Graph \(y_{1}\) and \(y_{2}\) in \([-5,5]\) by \([-2,10] .\) How many times do you think the two graphs cross? (b) Compare the corresponding changes in \(y_{1}\) and \(y_{2}\) as \(x\) changes from 1 to \(2,2\) to \(3,\) and so on. How large must \(x\) be for the changes in \(y_{2}\) to overtake the changes in \(y_{1} ?\) (c) Solve for \(x : x^{2}=2^{x}\) . \(\quad\) (d) Solve for \(x : x^{2}<2^{x}\)
Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75\(\%\) compounded annually.
Enter \(y_{1}=\sqrt{x}, y_{2}=\sqrt{1-x}\) and \(y_{3}=y_{1}+y_{2}\) on your grapher. (a) Graph \(y_{3}\) in \([-3,3]\) by \([-1,3]\) (b) Compare the domain of the graph of \(y_{3}\) with the domains of the graphs of \(y_{1}\) and \(y_{2}\) . (c) Replace \(y_{3}\) by \(y_{1}-y_{2}, \quad y_{2}-y_{1}, \quad y_{1} \cdot y_{2}, \quad y_{1} / y_{2}, \quad\) and \(\quad y_{2} / y_{1}\) in turn, and repeat the comparison of part (b). (d) Based on your observations in \((b)\) and \((c),\) what would you conjecture about the domains of sums, differences, products, and quotients of functions?
In Exercises \(25-26,\) show that the function is one-to-one, and graph its inverse. $$y=\tan x\( \)-\frac{\pi}{2}\( \)< x <$$\frac{\pi}{2}$$
In Exercises \(63-66,\) (a) graph \(f \circ g\) and \(g \circ f\) and make a conjecture about the domain and range of each function. (b) Then confirm your conjectures by finding formulas for \(f \circ g\) and \(g \circ f\) . $$f(x)=x-7, \quad g(x)=\sqrt{x}$$
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