Problem 76
Find all the inverses associated with the following functions and state their domains. $$f(x)=(x-4)^{2}$$
Problem 76
a. Find the linear function \(C=f(F)\) that gives the reading on the Celsius temperature scale corresponding to a reading on the Fahrenheit scale. Use the facts that \(C=0\) when \(F=32\) (freezing point) and \(C=100\) when \(F=212\) (boiling point). b. At what temperature are the Celsius and Fahrenheit readings equal?
Problem 77
Find all the inverses associated with the following functions and state their domains. $$f(x)=2 /\left(x^{2}+2\right)$$
Problem 77
Right-triangle relationships Use a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\sec ^{-1} x\right)$$
Problem 77
Automobile lease vs. purchase \(A\) car dealer offers a purchase option and a lease option on all new cars. Suppose you are interested in a car that can be bought outright for \(\$ 25,000\) or leased for a start-up fee of \(\$ 1200\) plus monthly payments of \(\$ 350\). a. Find the linear function \(y=f(m)\) that gives the total amount you have paid on the lease option after \(m\) months. b. With the lease option, after a 48 -month (4-year) term, the car has a residual value of \(\$ 10,000,\) which is the amount that you could pay to purchase the car. Assuming no other costs, should you lease or buy?
Problem 78
Right-triangle relationships Use a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cot \left(\tan ^{-1} 2 x\right)$$
Problem 78
Find all the inverses associated with the following functions and state their domains. $$f(x)=2 x /(x+2)$$
Problem 78
Surface area of a sphere The surface area of a sphere of radius \(r\) is \(S=4 \pi r^{2} .\) Solve for \(r\) in terms of \(S\) and graph the radius function for \(S \geq 0\).
Problem 79
A culture of bacteria has a population of 150 cells when it is first observed. The population doubles every 12 hr, which means its population is governed by the function \(p(t)=150 \cdot 2^{t / 12},\) where \(t\) is the number of hours after the first observation. a. Verify that \(p(0)=150,\) as claimed. b. Show that the population doubles every \(12 \mathrm{hr}\), as claimed. c. What is the population 4 days after the first observation? d. How long does it take the population to triple in size? e. How long does it take the population to reach \(10,000 ?\)
Problem 79
Right-triangle relationships Use a right triangle to simplify the given expressions. Assume \(x>0.\) $$\sin \left(\sec ^{-1}\left(\frac{\sqrt{x^{2}+16}}{4}\right)\right)$$
