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Chapter 0: Problem 24
Solve the equation and check your solution. $$ \frac{3 x}{2}+\frac{1}{4}(x-2)=10 $$
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In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=3 x+5 $$
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 \(f^{-1}\). $$ f(x)=x^{3}+1 $$
The numbers of households \(f\) (in thousands) in the United States from 1995 to 2003 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Census Bureau) $$ \begin{array}{|c|c|} \hline \text { Year, } t & \text { Households, } f(t) \\ \hline 5 & 98,990 \\ 6 & 99,627 \\ 7 & 101,018 \\ 8 & 102,528 \\ 9 & 103,874 \\ 10 & 104,705 \\ 11 & 108,209 \\ 12 & 109,297 \\ 13 & 111,278 \\ \hline \end{array} $$ (a) Find \(f^{-1}(108,209)\). (b) What does \(f^{-1}\) mean in the context of the problem? (c) Use the regression feature of a graphing utility to find a linear model for the data, \(y=m x+b\). (Round \(m\) and \(b\) to two decimal places.) (d) Algebraically find the inverse function of the linear model in part (c). (e) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(117,022)\). (f) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(108,209)\). How does this value compare with the original data shown in the table?
Average Price The average prices \(p\) (in thousands of dollars) of a new mobile home in the United States from 1990 to 2002 (see figure) can be approximated by the model $$ p(t)= \begin{cases}0.182 t^{2}+0.57 t+27.3, & 0 \leq t \leq 7 \\ 2.50 t+21.3, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=0\) corresponding to 1990. Use this model to find the average price of a mobile home in each year from 1990 to 2002 . (Source: U.S. Census Bureau)
Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ h(x)=x^{3}-5 $$
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