513 Domain of a Function (Summary) To find the domain of a function, given the equation that defines the function, remember that the value of x input into the equation must yield a real number for y when the function is evaluated. For the functions studied so far in this book, there are three cases to consider when determining domains. ■ Domain of a Function (Summary) ■ Equation Defining y as a Function of x 55. Electricity Consumption Suppose that in a certain area the consumption of electricity has increased at a continuous rate of 6% per year. If it continued to increase at this rate, find the number of years, to the nearest tenth, before twice as much electricity would be needed. 56. Electricity Consumption Suppose a conservation campaign, together with higher rates, causes demand for electricity to increase at a continuous rate of 2% per year. Find the number of years, to the nearest tenth, before twice as much electricity would be needed. (Modeling) Solve each problem that uses a logistic function. 57. Heart Disease As age increases, so does the likelihood of coronary heart disease (CHD). The fraction of people x years old with some CHD is modeled by ƒ1x2 = 0.9 1 + 271e-0.122x . (Data from Hosmer, D., and S. Lemeshow, Applied Logistic Regression, John Wiley and Sons.) (a) Evaluate ƒ1252 and ƒ1652 to the nearest hundredth. Interpret the results. (b) At what age, to the nearest year, does this likelihood equal 50%? 58. Tree Growth The height of a certain tree in feet after x years is modeled by ƒ1x2 = 50 1 + 47.5e-0.22x . (a) Make a table for ƒ starting at x = 10, and incrementing by 10. What appears to be the maximum height of the tree? (b) Graph ƒ and identify the horizontal asymptote. Explain its significance. (c) After how many years was the tree 30 ft tall? Round to the nearest tenth. Summary Exercises on Functions: Domains and Defining Equations Guidelines for Domain Restrictions 1. No input value can lead to 0 in a denominator, because division by 0 is undefined. 2. No input value can lead to an even root of a negative number, because this situation does not yield a real number. 3. No input value can lead to the logarithm of a negative number or 0, because this situation does not yield a real number. Summary Exercises on Functions: Domains and Defining Equations
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