§ 2.3 The Algebra of Functions – Finding the Domain Domain of a Function 117 Finding a Function’s Domain If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero. Exclude from a function’s domain real numbers that result in a square root of a negative number. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.3 Domain of a Function 117 • Consider the function 1 f ( x) x5 Because division by 0 is undefined (and not a real number), the denominator, x – 5, cannot be 0. Then x cannot be 5, and 5 is not in the domain of the function. Domainof f x | x is a real number and x 5 Blitzer, Intermediate Algebra, 5e – Slide #3 Section 2.3 Domain of a Function 117 • Now consider the function: g ( x) x7 The equation tells us to take the square root of x – 7. Because only nonnegative numbers have square roots that are real numbers, the expression under the square root must be nonnegative. Then x must be greater than or equal to 7. Domainof g x | x is a real number and x 7 Blitzer, Intermediate Algebra, 5e – Slide #4 Section 2.3 Domain of a Function 118 EXAMPLE 2 Find the domain of the function: f x 4x 2x 7 . SOLUTION Since the function f has no denominator or square root, there are no real numbers that when plugged into the function for x would cause the value of the function to yield something other than a real number. Therefore, the domain is: Domainof f x | x is a real number This is set notation and it is read: “the set of all x such that x is a real number.”. Using this notation, the rule stating the conditions for x follows the vertical bar which just means “such that.” Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.3 Domain of a Function 118 EXAMPLE Find the domain of the function: f x 2 6 x4 5 x . SOLUTION The function has no square roots so we don’t have to worry about pursuing that avenue. However the function does have x in two different denominators. Therefore I do the following: Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.3 Domain of a Function 118 f x CONTINUED 2 6 x4 5 x x40 x4 Set a denominator equal to zero Solve 5 x 0 x 5 Set a denominator equal to zero Solve Therefore, a denominator of the function is equal to zero when x = 4 or x = -5. Then the domain is: Domainof f x | x is a real number and x 4 and x 5 Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.3 Domain of a Function 118 Check Point 1a Find the domain of the function: 1 f x x 3. 2 SOLUTION Since the function f has no denominator or square root, there are no real numbers that when plugged into the function for x would cause the value of the function to yield something other than a real number. Therefore, the domain is: Domainof f x | x is a real number Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.3 Domain of a Function 117-118 Check Point 1b Find the domain of the function: 7 x .4 g x x5 Because division by 0 is undefined (and not a real number), the denominator, x + 5, cannot be 0. Then x cannot be -5, and -5 is not in the domain of the function. Domainof g x | x is a real number and x - 5 Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.3 DONE