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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

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Jul 26, 2012
Find the Domain
by: Staff

The answer:

Part I

Domain: 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 undefined

2. 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 computed

3. a logarithm can only be computed for a positive number which is greater than zero

4. 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) = 0

3x - 10 + 10 = 0 + 10

3x + 0 = 10

3x = 10

3x / 3 = 10 / 3

x * (3 / 3) = 10 / 3

x * (1) = 10 / 3

x = 10 / 3

x = 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 ≥ 0

7x + 5 - 5 ≥ 0 - 5

7x + 0 ≥ 0 - 5

7x ≥ - 5

7x / 7 ≥ - 5 / 7

x * (7 / 7) ≥ - 5 / 7

x * (1) ≥ - 5 / 7

x ≥ - 5 / 7

x 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) = 0

3x - 5 + 5 = 0 + 5

3x + 0 = 5

3x = 5

3x / 3 = 5 / 3

x * (3 / 3) = 5 / 3

x * (1) = 5 / 3

x = 5 / 3

x = 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.

 Math – graph of  y = (9x² - 25) / (3x - 5)



Math – graph of  y = (3x + 5)





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

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