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Chapter 0: Problem 42
Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{1}{x-2}+\frac{3}{x+3}=\frac{4}{x^{2}+x-6} $$
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Geometry A rectangle is bounded by the \(x\)-axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x\), and determine the domain of the function.
In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{8 x-4}{2 x+6} $$
Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-x^{3}+6 x^{2}+x &\quad x_{1}=1, x_{2}=6 \end{array} $$
Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ f(x)=x \sqrt{1-x^{2}} $$
In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\sqrt{2 x+3} $$
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