Section 3.4 The Domain and Range of Functions 193 Student Check 2 Find the domain and range of each function. a. b. c. 2 2 4 4 –2 –4 –2 –4 x y 2 y 2 4 4 –2 –4 –2 –4 x 2 2 4 4 –2 –4 –2 –4 x y Using an Equation to Find Its Domain To find the domain of a function given its equation, we can graph the equation and read the graph to determine its domain. This, however, is not necessary as we can determine the domain of a function from its equation. When examining an equation to find its domain, the goal is to determine what values of x make the function defined. A function is defined at a value of x if the corresponding y-value is a real number. If there are any values of x that make the function undefined or not a real number, they must be excluded from the domain of the function. Definition: The domain of a function represented algebraically is the set of real numbers, excluding any number that makes the equation undefined or not a real number. Recall that division by zero is undefined and that the square root of a negative number is not a real number. If a function involves a fraction with a variable in the denominator or involves the square root of an algebraic expression, the function may have values that make it undefined or not real. Examples of these types of functions are f(x) = 1 x + 5 g(x) = 1x + 3 The function f (x) is undefined when the denominator is equal to zero, or when x + 5 = 0 x=-5 So, the domain of f(x) is all real numbers except x=-5. We can write this as (-∞, -5) ∪ (-5, ∞). The function g(x) is not a real number when the expression inside the square root is negative. So, as long as the expression inside the square root is positive or zero, the function is defined. We must solve the following inequality to determine the domain of g(x). x 1 3 $ 0 x $ 23 So, the domain of g(x) is -3, ∞). Procedure: Determining the Domain of a Function from Its Equation Step 1: If the function does not contain a fraction with a variable in the denominator or the square root of an expression, the domain is all real numbers. Objective 3 ▶ Find the domain of a function given an equation.
hendricks_intermediate_algebra_1e_ch1_3
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