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Chapter 1: Problem 83
Determine whether the statement is true or false. Justify your answer. The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.
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The simple interest on an investment is directly proportional to the amount of the investment. By investing \(\$ 3250\) in a certain bond issue, you obtained an interest payment of \(\$ 113.75\) after 1 year. Find a mathematical model that gives the interest \(I\) for this bond issue after 1 year in terms of the amount invested \(P\)
Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.
On a yardstick with scales in inches and centimeters, you notice that 13 inches is approximately the same length as 33 centimeters. Use this information to find a mathematical model that relates centimeters \(y\) to inches \(x\). Then use the model to find the numbers of centimeters in 10 inches and 20 inches.
The function given by \(f(x)=k\left(2-x-x^{3}\right)\) has an inverse function, and \(f^{-1}(3)=-2 .\) Find \(k\).
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.
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