514 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Unless otherwise specified, we determine domains as follows. • The domain of a polynomial function is the set of all real numbers. • The domain of an absolute value function is the set of all real numbers for which the expression inside the absolute value bars (the argument) is defined. • If a function is defined by a rational expression, the domain is the set of all real numbers for which the denominator is not 0. • The domain of a function defined by a radical with even root index is the set of all real numbers that make the radicand greater than or equal to 0. If the root index is odd, the domain is the set of all real numbers for which the radicand is itself a real number. • For an exponential function with constant base, the domain is the set of all real numbers for which the exponent is a real number. • For a logarithmic function, the domain is the set of all real numbers that make the argument of the logarithm greater than 0. Equation Defining y as a Function of x For y to be a function of x, it is necessary that every input value of x in the domain leads to one and only one value of y. To determine whether an equation such as x - y3 = 0 or x - y2 = 0 represents a function, solve the equation for y. In the first equation above, doing so leads to y = 23 x. Notice that every value of x in the domain (that is, all real numbers) leads to one and only one value of y. So in the first equation, we can write y as a function of x. However, in the second equation above, solving for y leads to y = {2x. If we let x = 4, for example, we obtain two values of y: -2 and 2. Thus, in the second equation, we cannot write y as a function of x. EXERCISES Find the domain of each function. Write answers using interval notation. 1. ƒ1x2 = 3x - 6 2. ƒ1x2 = 22x - 7 3. ƒ1x2 = 0 x + 40 4. ƒ1x2 = x + 2 x - 6 5. ƒ1x2 = -2 x2 + 7 6. ƒ1x2 = 2x2 - 9 7. ƒ1x2 = x2 + 7 x2 - 9 8. ƒ1x2 = 23 x3 + 7x - 4 9. ƒ1x2 = log 5116 - x 22 10. ƒ1x2 = log x + 7 x - 3 11. ƒ1x2 = 2x2 - 7x - 8 12. ƒ1x2 = 21/x 13. ƒ1x2 = 1 2x2 - x + 7 14. ƒ1x2 = x2 - 25 x + 5 15. ƒ1x2 = 2x3 - 1
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