# Find the domain of the following functions

Find the domain of the following functions

(i) y = 15 - 2x

(ii) y = 14 / (3x - 10)

(iii) y = √(7x + 5)

(iv) y = (9x² - 25) / (3x - 5)

(v) y = ln(4x - 5)

### Comments for Find the domain of the following functions

 Jul 26, 2012 Find the Domain by: Staff The answer: Part IDomain: the domain of a function is the set of every valid input value (for your problems, every valid x value).What input values are not valid?1. input values which cause a division by zero (0), since dividing by zero is undefined2. input values which cause the discriminant of an even root (such as a square root) to be negative, since the square root of a negative number cannot be computed3. a logarithm can only be computed for a positive number which is greater than zero4. input values which are outside any conditional limitations. (i) y = 15 - 2x >>> the final answer:Domain: = {x | x ∈ ℝ} The domain for y = 15 - 2x is the set of all real numbers, ℝ.(ii) y = 14 / (3x - 10) Some values of x will cause a division by zero (0)(3x - 10) = 03x - 10 + 10 = 0 + 103x + 0 = 103x = 103x / 3 = 10 / 3x * (3 / 3) = 10 / 3x * (1) = 10 / 3x = 10 / 3x = 3 1/3 when x = 3 1/3, (3x - 10) = 0; x = 3 1/3 is not part of the domain>>> the final answer:Domain in Set Builder Format:Domain: = {x | x ∈ ℝ, x ≠ 3 1/3}Or Domain in INTERVAL Notation:Domain: (-∞,3 1/3) ∪(3 1/3, ∞)(iii) y = √(7x + 5) The expression 7x + 5 must be ≥ 0 for the square root to be valid (the discriminant of the square root must be ≥ 0)7x + 5 ≥ 07x + 5 - 5 ≥ 0 - 57x + 0 ≥ 0 - 57x ≥ - 57x / 7 ≥ - 5 / 7x * (7 / 7) ≥ - 5 / 7x * (1) ≥ - 5 / 7x ≥ - 5 / 7x must be greater than or equal to - 5 / 7 for the square root function to be valid. When x is less than - 5 / 7, (7x + 5) becomes a negative number. The square root of a negative number cannot be computed.Any value of x < - 5 / 7 is not part of the domain>>> the final answer:Domain in Set Builder Format:Domain: = {x | x ∈ ℝ, x ≥ - 5 / 7 }Or Domain in INTERVAL Notation:Domain: [- 5 / 7, ∞)(iv) y = (9x² - 25) / (3x - 5) Some input values of x will cause a division by zero (0)(3x - 5) = 03x - 5 + 5 = 0 + 53x + 0 = 53x = 53x / 3 = 5 / 3x * (3 / 3) = 5 / 3x * (1) = 5 / 3x = 5 / 3x = 1 2/3 when x = 1 2/3, (3x - 5) = 0; x = 1 2/3 is not part of the domain>>> the final answer:Domain in Set Builder Format:Domain: = {x | x ∈ ℝ, x ≠ 1 2/3 }Or Domain in INTERVAL Notation:Domain: (-∞,5/3)∪(5/3,+∞)------------------------------------

 Jul 26, 2012 Find the Domain by: Staff ------------------------------------ Part II (v) y = ln(4x - 5) y = ln(4x - 5) For a natural log to be valid, the expression 4x - 5 must be greater than zero (4x - 5) > 0 4x - 5 + 5 > 0 + 5 4x - 5 + 5 > 0 + 5 4x + 0 > 5 4x > 5 4x / 4 > 5 / 5 x * (4 / 4) > 5 / 4 x * (1) > 5 / 4 x > 5 / 4 x must be greater than 5 / 4 for any logarithm to be computed for the expression (4x - 5). When x is less than or equal to 5 / 4, no logarithm (regardless of the number base) can be computed. Any value of x ≤ 5 / 4 is not part of the domain >>> the final answer: Domain in Set Builder Format: Domain: = {x | x ∈ ℝ, x > 5 / 4} Or Domain in INTERVAL Notation: Domain: (5 / 4, ∞) Thanks for writing. Staff www.solving-math-problems.com

 Aug 26, 2012 Need to expand first by: Anonymous I think question for iv answer should be (3x-5)(3x+5)/3x-5 (need to expand first) therefore :- 3x+5 = 0 3x+5-5 = 0-5 3x = -5 x = -5/3

 Aug 26, 2012 Find the domain of function iv by: Staff Hello Anonymous,Thanks for writing.Your comments bring out three very important points:    1. The equation you simplified and solved, 0 = (9x² - 25) / (3x - 5), is not the same as the equation which was given in the problem statement.       The equation given in the problem statement is: y = (9x² - 25) / (3x - 5)    2. The value of the variable “y” is not necessarily equal to 0.     3. The equation given in the problem statement, y = (9x² - 25) / (3x - 5), should not be simplified to y = 3x + 5 before determining the domain of “x” values.       The function f(x) = (9x² - 25) / (3x - 5) and it’s simplified version f(x) = 3x + 5 are completely different from one another. Thanks for writing.Staff www.solving-math-problems.com

 Sep 14, 2012 Still confused by: Anonymous Dear Staff, Can this answer is y=3x+5 instead of finding the value of x? Rgds

 Sep 15, 2012 domain by: Staff Hello Anonymous, Finding the domain means to find all the possible values of “x” which can be used in the function. You cannot simply the function. The domain of the function y = (9x² - 25) / (3x - 5) is: The set of all real numbers which the exception of 5/3. If x = 5/3, the denominator of the fraction (9x² - 25) / (3x - 5) would be zero. y = (9x² - 25) / (3x - 5) y = (9*(5/3)² - 25) / (3*(5/3) - 5) y = (9*(5/3)² - 25) / (5 - 5) y = (9*(5/3)² - 25) / (0) Dividing by zero is undefined. That is why the domain is the set of all real numbers with the exception of 5/3. Thanks for writing. Staff www.solving-math-problems.com

 Sep 16, 2012 Thank you by: Anonymous Dear Staff, Thanks for the clarification. Understood now. Regards