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Chapter 1 | Functions and Graphs
which 4−3 x ≥0. Therefore, the domain is { x | x ≤4/3}. d. The cube root is defined for all real numbers, so the domain is the interval (−∞, ∞).
Find the domain for each of the following functions: f ( x ) = (5−2 x )/( x 2 +2) and g ( x ) = 5 x −1.
1.12
Transcendental Functions Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are sin x , cos x , tan x , cot x , sec x , andcsc x . (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form f ( x ) = b x , where the base b >0, b ≠1. A logarithmic function is a function of the form f ( x ) = log b ( x ) for some constant b >0, b ≠1, where log b ( x ) = y if and only if b y = x . (We also discuss exponential and logarithmic functions later in the chapter.) Example 1.18 Classifying Algebraic and Transcendental Functions
Classify each of the following functions, a. through c., as algebraic or transcendental. a. f ( x ) = x 3 +1 4 x +2 b. f ( x ) =2 x 2 c. f ( x ) = sin(2 x )
Solution a. Since this function involves basic algebraic operations only, it is an algebraic function. b. This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.) c. As in part b., this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.
Is f ( x ) = x /2 an algebraic or a transcendental function?
1.13
Piecewise-Defined Functions Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is knownas a piecewise-defined function . The absolute value function is an example of a piecewise-defined function because
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