Chapter 0: Problem 48
Find an equation of the line satisfying the conditions. Write your answer in the slope-intercept form. Passes through \((-3,3)\) and has slope 0
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Show that \(f\) and \(g\) are inverses of each other by verifying that \(f[g(x)]=x\) and \(g[f(x)]=x\). $$ \begin{aligned} &f(x)=4(x+1)^{2 / 3}, \text { where } x \geq-1 \\ &g(x)=\frac{1}{8}\left(x^{3 / 2}-8\right), \text { where } x \geq 0 \end{aligned} $$
You are given the graph of a function \(f .\) Determine whether \(f\) is one-to- one.
Find the exact value of the given expression. $$ \tan ^{-1} \sqrt{3} $$
Let \(f(x)=2 x^{3}-5 x^{2}+x-2\) and \(g(x)=2 x^{3}\). a. Plot the graph of \(f\) and \(g\) using the same viewing window: \([-5,5] \times[-5,5]\). b. Plot the graph of \(f\) and \(g\) using the same viewing window: \([-50,50] \times[-100,000,100,000] .\) c. Explain why the graphs of \(f\) and \(g\) that you obtained in part (b) seem to coalesce as \(x\) increases or decreases without bound. Hint: Write \(f(x)=2 x^{3}\left(1-\frac{5}{2 x}+\frac{1}{2 x^{2}}-\frac{1}{x^{3}}\right)\) and study its behavior for large values of \(x\).
Classify each function as a polynomial function (state its degree), a power function, a rational function, an algebraic function, a trigonometric function, or other. a. \(f(x)=2 x^{3}-3 x^{2}+x-4\) b. \(f(x)=\sqrt[3]{x^{2}}\) c. \(g(x)=\frac{x}{x^{2}-4}\) d. \(f(t)=3 t^{-2}-2 t^{-1}+4\) e. \(h(x)=\frac{\sqrt{x}+1}{\sqrt{x}-1}\) f. \(f(x)=\sin x+\cos x\)
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