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Thread: Specify the Domain

  1. #1
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    Specify the Domain

    Specify the domain of each variable.

    1. ax + b

    Here, x is the variable. I say a and b are constants.
    Therefore, the domain is ALL REAL NUMBERS.
    Is this right?

    2. ax^(1/3) + b

    Again, x is the variable. I know that a and b are constants.
    I decided to rewrite question 2 as a*cuberoot (x) + b. Is this correct? What would be the domain in this case?
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    Re: Specify the Domain

    1. ax + b is linear , and hence domain is all real x and range all real y

    2. $x^{\frac13}=\sqrt[3]{x}$, so yes, that is a correct rewrite

    if you consider the basic cube root curve $y=\sqrt[3]{x}$ its domain is all real x and its range is all real y.
    by multiplying it by a, you are essentially only affecting the 'narrowness' or 'width' of the curve, and adding b will move it up or down.
    neither of these operations will have any impact on the domain or range of the cubic
    Last edited by jacs; Jan 4th 2017 at 11:35 PM. Reason: typo
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    Re: Specify the Domain

    Quote Originally Posted by jacs View Post
    1. ax + b is linear , and hence domain is all real x and range all real y

    2. $x^{\frac13}=\sqrt[3]{x}$, so yes, that is a correct rewrite

    if you consider the basic cube root curve $y=\sqrt[3]{x}$ its domain is all real x and its range is all real y.
    by multiplying it by a, you are essentially only affecting the 'narrowness' or 'width' of the curve, and adding b will move it up or down.
    neither of these operations will have any impact on the domain or range of the cubic
    Can you graph y = cuberoot (x) and show me how to determine the domain and range from the graph itself?
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    Re: Specify the Domain

    Specify the Domain-cubic.jpg

    the blue curve is $y=x^3$

    the red curve is $y=\sqrt[3]x$

    you can see the the cube root curve is very similar in shape to the cubic and are actually inverse functions
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    Re: Specify the Domain

    To determine the domain for a function, f(x), look for values where f cannot be applied. Given f(x)= ax^(1/3)+ b, we can take the cube root of any number, we can multiply any number by a and we can add b. The domain is 'all numbers'. To determine the range, solve the equation for x. From y= ax^(1/3)+ b, ax^(1/3)= y- b, x^(1/3)= y/a+ b/a, and, finally, x= (y/a+ b/a)^3. For any value of y we can: divide by a, add b/a, and then cube. The range is "all numbers'.

    A more interesting function is y= x^2. we can, of course, square any number so the domain is 'all numbers'. But solving for x involves taking the square root of y. We can only take the square root (x and y are real numbers here) of non-negative numbers. The range is 'all non-negative numbers".
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    Re: Specify the Domain

    Quote Originally Posted by jacs View Post
    Click image for larger version. 

Name:	cubic.jpg 
Views:	2 
Size:	76.9 KB 
ID:	36835

    the blue curve is $y=x^3$

    the red curve is $y=\sqrt[3]x$

    you can see the the cube root curve is very similar in shape to the cubic and are actually inverse functions
    In what way does your picture answer my question concerning domain and range? What does your picture say about domain and range?
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    Re: Specify the Domain

    Quote Originally Posted by HallsofIvy View Post
    To determine the domain for a function, f(x), look for values where f cannot be applied. Given f(x)= ax^(1/3)+ b, we can take the cube root of any number, we can multiply any number by a and we can add b. The domain is 'all numbers'. To determine the range, solve the equation for x. From y= ax^(1/3)+ b, ax^(1/3)= y- b, x^(1/3)= y/a+ b/a, and, finally, x= (y/a+ b/a)^3. For any value of y we can: divide by a, add b/a, and then cube. The range is "all numbers'.

    A more interesting function is y= x^2. we can, of course, square any number so the domain is 'all numbers'. But solving for x involves taking the square root of y. We can only take the square root (x and y are real numbers here) of non-negative numbers. The range is 'all non-negative numbers".
    Informative reply.
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    Re: Specify the Domain

    Quote Originally Posted by mathdad1965 View Post
    In what way does your picture answer my question concerning domain and range? What does your picture say about domain and range?
    Looking at the red curve.... is there anything that x can't be? No, it is a continuous graph that extends infinitely from left to right. So there is no restriction on what x can be. Therefore the domain is "all real numbers".

    Similarly for y. As you look at the curve from top to bottom, y can be any real number. (Note: the graph extends upwards to the right and downwards to the left indefinitely) Hence the range is "all real numbers".
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    Re: Specify the Domain

    Quote Originally Posted by Debsta View Post
    Looking at the red curve.... is there anything that x can't be? No, it is a continuous graph that extends infinitely from left to right. So there is no restriction on what x can be. Therefore the domain is "all real numbers".

    Similarly for y. As you look at the curve from top to bottom, y can be any real number. (Note: the graph extends upwards to the right and downwards to the left indefinitely) Hence the range is "all real numbers".
    Now I get it. All real numbers can be written as
    (-infinity, infinity) in interval notation, right?
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    Re: Specify the Domain

    Quote Originally Posted by mathdad1965 View Post
    Now I get it. All real numbers can be written as
    (-infinity, infinity) in interval notation, right?
    Yes that's right.
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    Re: Specify the Domain

    Good information here.
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