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Lecture Section 1 – 1 Domain When we state the domain for a function f (x) we are listing ALL the x values that we CAN put into the function and get a y value that is a real number. For many functions the domain of the function is ALL REAL NUMBERS. You CAN put any real number into the function for x and get a y value that is a real number. We write this domain as x ∈ Reals or x ∈ ℜ Example 1! Example 2 ! State the domain of State the domain of f(x) = sin(2x) 2 f(x) = 3x − 9x You CAN put any real number into 3x2 − 9 and get a y value that is a real number. x ∈ Reals or x ∈ ℜ ! You CAN put any real number into sin(2x) and get a y value that is a real number. x ∈ Reals or x ∈ ℜ ! Domain Restrictions There are functions where the domain of the function is NOT ALL REAL NUMBERS. When you put certain real number into the function for x you DO NOT get a y value that is a real number. Example 1! Example 2 ! 2x x−5 You CANNOT put x = 5 into f (x) and get a y value that is a real number.! 2(5) 10 f(5) = = 5−5 0 which is NOT a real number. You CANNOT put x = 3 into f (x) ! ! and get a y value that is a real number. Example 3! Example 4 ! f(x) = f(x) = f(3) = 3 - 4 = −1 which is NOT a real number. f(x) = log(x) f(x) = cot(x) You CANNOT put x = 0 into f (x) and get a y value that is a real number. Log(0) is undefined which is NOT a real number. ! . Lecture 1– 1Domain! x-4 You CANNOT put x = 0 into f (x) and get a y value that is a real number. cot(0) = 1 / 0 which is undefined which is NOT a real number. !! Page 1 of 5! © 2016 Eitel There are several common domain issues we encounter with functions in this class. The examples below will cover the 4 most common domain issues and discuss how to state the domain for each type of function. Rational Functions g(x) A. f (x) = f (x) has a domain restriction at all the x values where the denominator is equal h(x) to 0. The domain for f (x) is all real x values EXCEPT the x values where h(x) = 0 Example 1! f (x) = Example 2 ! x+3 (x − 2)(x + 4) f (x) = x+3 (x − 5) 3 (3x − 2) set each factor in the demominator equal to 0 and solve for x set each factor in the demominator equal to 0 and solve for x x−2=0 x=2 x−5=0 x=5 x+4=0 x = −4 Domain { x ∈ Reals | x ≠ 2 or − 4} 3x − 2 = 0 x = 2/3 Domain { x ∈ Reals | x ≠ 5 or 2 / 3} ! ! Functions involving square roots B. f (x) = g(x) The square root function requires that the values under the square root must be greater than or equal to 0. The domain for f (x) is all real x values where g(x) ≥ 0 Example 1! Example 2 ! f (x) = 3x − 2 set the function under the f (x) = 4 − 5x sign set the function under the sign greater than or equal to 0 and solve for x greater than or equal to 0 and solve for x 3x − 2 ≥ 0 3x ≥ 2 x ≥ 2/3 4 − 5x ≥ 0 −5x ≥ −4 x ≤ 4 /5 Domain { x ∈ Reals | x ≥ 2 / 3} Domain { x ∈ Reals | x ≤ 4 / 5} Lecture 1– 1Domain! ! Page 2 of 5! © 2016 Eitel Example 3! f (x) = Example 4 ! x−2 x+3 set each function under a f (x) = sign −5 must be greater than or equal to 0 x−2 since − 5 is negitive x − 2 must be ≥ 0 x − 2 is in the denominator so x − 2 cannot equal to zero. greater than or equal to 0 and solve for x The domain is the INTERSECTION of the two inequalities x − 2 ≥ 0 and x + 3 ≥ 0 solve for x x≥2 and x ≥ −3 −5 x−2 ! x−2>0 solve for x x>2 the INTERSECTION of the two inequalities is x≥2 Domain { x ∈ Reals | x ≥ 2} Domain { x ∈ Reals | x > 2} Log Functions C. f (x) = log b ( g(x)) The log function requires that the values for g(x) must be greater than 0. The domain for f (x) is all real x values where g(x) > 0 The domain is the the INTERSECTION of the two inequaltites inequalities Example 1! Example 2 ! f (x) = log 3 (3x − 6) f (x) = log 3 (6 − 2x) set the function inside the ( ) greater than 0 and solve for x set the function inside the ( ) greater than 0 and solve for x x−6>0 3x > 6 x>2 6 − 2x > 0 −2x > −6 x<3 Domain { x ∈ Reals | x > 2} Domain { x ∈ Reals | x < 3} Lecture 1– 1Domain! ! Page 3 of 5! © 2016 Eitel Mixed Examples Example 1! f (x) = Example 2 ! log(x − 1) x+4 f (x) = x +1 log(x + 3) (x − 1) must be greater than 0 and x + 4 must be greater than 0 not equal to 0 as it is in the denominator (x − 1) must be greater than or equal to 0 and x + 3 must be greater than 0 The domain is the INTERSECTION of the two inequalities x − 1 > 0 and x + 4 > 0 solve for x x >1 and x > −4 The domain is the INTERSECTION of the two inequalities x + 1 ≥ 0 and x + 4 > 0 solve for x x ≥ −1 and x > −3 the INTERSECTION of the two inequalities is x >1 Domain { x ∈ Reals | x > 1} ! the INTERSECTION of the two inequalities is x ≥ −1 Domain { x ∈ Reals | x ≥ −1} Example 3! f (x) = Example 4 3− x log(x − 4) f (x) = 2− x (x − 3) (x + 4) (3 − x) must be greater than or equal to 0 and x − 4 must be greater than 0 (2 − x) must be greater than or equal to 0 and x − 3 ≠ 0 and x − 3 ≠ 0 The domain is the INTERSECTION of the two inequalities 3 − x ≥ 0 and x + 4 > 0 solve for x x ≤ −1 and x > −4 The domain is the INTERSECTION of the two inequalities x + 1 ≥ 0 and x + 4 > 0 solve for x x ≥ −1 and x > −3 the INTERSECTION of the two inequalities is −4 < x ≤ −1 Domain { x ∈ Reals | − 4 < x ≤ −1} ! the INTERSECTION of the two inequalities is x ≥ −1 Domain { x ∈ Reals | x ≥ −1} Lecture 1– 1Domain! Page 4 of 5! © 2016 Eitel Trigonometric Functions D. The domain of the 6 trigonometric functions must be examined for each function separately. f (x) = sin(x) f (x) = csc(x) f (x) = cos(x) Example 1! f (x) = sec(x) f (x) = tan(x) f (x) = cot (x) Example 2 ! f (x) = cos(x) f (x) = sin(x) cos(x) is defined for all Real numbers cos(x) is defined for all Real numbers Domain: { x ∈ Reals } Domain: { x ∈ Reals } Example 3! ! ! ! Example 4 ! f (x) = cot(x) f (x) = tan(x) cot (x)is undefined for the following values tan(x)is undefined for the following values ! −5π −3π −π π 3π 5π ...... , , , , , ..... 2 2 2 2 2 2 Domain: π { x ∈ Reals | x ≠ + π • k} k ∈integers 2 Example 5! Example 6 ! f (x) = csc (x) f (x) = sec (x) csc (x)is undefined for the following values tan(x)is undefined for the following values ! −5π −3π −π π 3π 5π ...... , , , , , ..... 2 2 2 2 2 2 Domain: π { x ∈ Reals | x ≠ + π • k} k ∈integers 2 Lecture 1– 1Domain! ...... − 2π ,− 2π ,− π ,0, π , 2π , 3π ..... Domain: { x ∈ Reals | x ≠ π + π • k} k ∈integers Page 5 of 5! ...... − 2π ,− 2π ,− π ,0, π , 2π , 3π ..... Domain: { x ∈ Reals | x ≠ π + π • k} k ∈integers © 2016 Eitel