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Chapter 1: Problem 54
Let \(g(x)=x^{2}+3 .\) Find a function \(f\) that produces the given composition. $$(g \circ f)(x)=x^{2 / 3}+3$$
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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\sin (a+b)=\sin a+\sin b\) b. The equation \(\cos \theta=2\) has multiple real solutions. c. The equation \(\sin \theta=\frac{1}{2}\) has exactly one solution. d. The function \(\sin (\pi x / 12)\) has a period of 12 e. Of the six basic trigonometric functions, only tangent and cotangent have a range of \((-\infty, \infty)\) f. \(\frac{\sin ^{-1} x}{\cos ^{-1} x}=\tan ^{-1} x\) g. \(\cos ^{-1}(\cos (15 \pi / 16))=15 \pi / 16\) h. \(\sin ^{-1} x=1 / \sin x\)
Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. $$\sin ^{-1}(-1)$$
Right-triangle relationships Use a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\tan ^{-1}\left(\frac{x}{\sqrt{9-x^{2}}}\right)\right)$$
Without using a graphing utility, sketch the graph of \(y=2^{x} .\) Then on the same set of axes, sketch the graphs of \(y=2^{-x}, y=2^{x-1}, y=2^{x}+1,\) and \(y=2^{2 x}\)
Square wave Graph the square wave defined by $$f(x)=\left\\{\begin{array}{ll}0 & \text { if } x<0 \\\1 & \text { if } 0 \leq x<1 \\\0 & \text { if } 1 \leq x<2 \\\1 & \text { if } 2 \leq x<3\end{array}\right.$$
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