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Chapter 1: Problem 31
Find the zeros of the function algebraically. $$ f(x)=\sqrt{2 x}-1 $$
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Determine whether the function has an inverse function. If it does, find the inverse function. $$ q(x)=(x-5)^{2} $$
The lengths (in feet) of the winning men's discus throws in the Olympics from 1920 through 2008 are listed below. (Source: International Olympic Committee) $$\begin{array}{llllll} 1920 & 146.6 & 1956 & 184.9 & 1984 & 218.5 \\ 1924 & 151.3 & 1960 & 194.2 & 1988 & 225.8 \\ 1928 & 155.3 & 1964 & 200.1 & 1992 & 213.7 \\ 1932 & 162.3 & 1968 & 212.5 & 1996 & 227.7 \\ 1936 & 165.6 & 1972 & 211.3 & 2000 & 227.3 \\ 1948 & 173.2 & 1976 & 221.5 & 2004 & 229.3 \\ 1952 & 180.5 & 1980 & 218.7 & 2008 & 225.8 \end{array}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the length of the winning discus throw (in feet) and let \(t=20\) represent 1920 (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men's discus throw in the year 2012 .
Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. An overhead garage door has two springs, one on each side of the door (see figure). A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door by the springs when the door is closed.
Use the given value of \(k\) to complete the table for the direct variation model $$y=k x^{2}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=k x^{2} & & & & & \\ \hline \end{array}$$ $$ k=1 $$
Determine whether the function has an inverse function. If it does, find the inverse function. $$ h(x)=-\frac{4}{x^{2}} $$
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