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Find a mathematical model for the verbal statement. \(y\) varies inversely as the square of \(x\).

Find all real values of \(x\) such that \(f(x)=0\). $$f(x)=x^{3}-x$$

During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of \(0.5\) inch per hour for the final hour. Write and graph a piecewise- defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm?

Find a mathematical model for the verbal statement. The rate of change \(R\) of the temperature of an object is directly proportional to the difference between the temperature \(T\) of the object and the temperature \(T_{e}\) of the environment in which the object is placed.

Approximating Relative Minima or Maxima Use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. $$f(x)=-x^{2}+3 x-2$$

Find a mathematical model for the verbal statement. For a constant temperature, the pressure \(P\) of a gas is inversely proportional to the volume \(V\) of the gas.

Determine whether the statement is true or false. Justify your answer. A piecewise-defined function will always have at least one \(x\) -intercept or at least one \(y\) -intercept.

Write an equation for the function described by the given characteristics. The shape of \(f(x)=x^{3},\) but shifted 13 units to the right

Use the Midpoint Formula three times to find the three points that divide the line segment joining \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) into four parts.

Plot the points (2,1),(-3,5) and (7,-3) on a rectangular coordinate system. Then change the signs of the indicated coordinates of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs. (a) The sign of the \(x\)-coordinate is changed. (b) The sign of the \(y\)-coordinate is changed. (c) The signs of both the \(x\)- and \(y\)-coordinates are changed.