91Ó°ÊÓ

Solve for \(x\) without using a calculating utility. Use the natural logarithm anywhere that logarithms are needed. $$ 3 e^{-2 x}=5 $$

Determine whether the statement is true or false. Explain your answer. If a curve passes through the point \((2,6)\) and \(y\) is inversely proportional to \(x,\) then the constant of proportionality is \(3 .\)

Solve for \(x\) without using a calculating utility. Use the natural logarithm anywhere that logarithms are needed. $$ 2 e^{3 x}=7 $$

Express the function in piecewise form without using absolute values. [Suggestion: It may help to generate the graph of the function.] $$ \text { (a) } f(x)=|x|+3 x+1 \quad \text { (b) } g(x)=|x|+|x-1| $$

Find formulas for \(f+g, f-g, f g,\) and \(f / g,\) and state the domains of the functions. $$ f(x)=2 \sqrt{x-1}, g(x)=\sqrt{x-1} $$

Let \(f(x)=2 x^{3}+5 x+3 .\) Find \(x\) if \(f^{-1}(x)=1\)

Solve for \(x\) without using a calculating utility. Use the natural logarithm anywhere that logarithms are needed. $$ e^{x}-2 x e^{x}=0 $$

Express the function in piecewise form without using absolute values. [Suggestion: It may help to generate the graph of the function.] $$ \text { (a) } f(x)=3+|2 x-5| \quad \text { (b) } g(x)=3|x-2|-|x+1| $$

Find formulas for \(f+g, f-g, f g,\) and \(f / g,\) and state the domains of the functions. $$ f(x)=\frac{x}{1+x^{2}}, g(x)=\frac{1}{x} $$

Determine whether the statement is true or false. Explain your answer. Curves in the family \(y=-5 \sin (A \pi x)\) have amplitude 5 and period \(2 /|A| .\)