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Math Help - Explain Domain and range of function f(x)

  1. #1
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    Angry Explain Domain and range of function f(x)

    Determine the range and function defined as y=sqrt (25-x^2)
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  2. #2
    Senior Member nikhil's Avatar
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    y= <br />
\sqrt{25-x^2}<br />
    for domain
    25-x^2>=0
    x^2<=25
    therefor x belongs to [-5,5]
    hence domain is [-5,5].
    Since its minimum value is 0 and maximum 5 and the function is continuous in its domain so its range is [0,5]
    note:I hope you know what is range and what is domain.if you have a doubt,you may ask it.
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  3. #3
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    Quote Originally Posted by Ngigi View Post
    Determine the range and function defined as y=sqrt (25-x^2)
    The graph is the top half of a circle of radius 5 and centre at the origin:

    y = \sqrt{25 - x^2} \Rightarrow y^2 = 25 - x^2 \Rightarrow x^2 + y^2 = 5^2.

    From this graph it should be clear what the domain and range are .....
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  4. #4
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    The Domain of a given function is the set of 'input' values (In this case , the x values) for which the function is defined.

    The function, y=\sqrt{(25-x^2)}, is a composed square root function. The domain is found by solving the inequality 25-x^2 \ge 0.

    25-x^2 \ge 0
    -x^2 \ge -25
    x^2 \le 25
    x^2\leq 25 \implies \sqrt{x^2}\leq 5

    Therefore the domain will be [-5, 5].
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    The Range of a given function is the set of 'output' values (In this case , the y values) for the function.

    The function, y=\sqrt{(25-x^2)}, is a quadratic hence \pm x values will give the same range. (This is a many to one function)

    When x=0 \implies y=\sqrt{ (25-(0)^2)} = 5 . When x=\pm 5 \implies y = \sqrt{ (25-(\pm 5)^2)} = 0.

    Therefore the range is [0,5]
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    Visually, when a graph is drawn, the domain and range can be seen.
    Attached Thumbnails Attached Thumbnails Explain Domain and range of function f(x)-diagram.jpg  
    Last edited by Simplicity; July 3rd 2008 at 07:50 AM. Reason: Wording Correction
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  5. #5
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by Air View Post
    [...]
    x^2 \le 25
    x \le \pm 5
    [...]
    There aren't many x such that x\leq \pm 5
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  6. #6
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    Quote Originally Posted by flyingsquirrel View Post
    There aren't many x such that x\leq \pm 5
    I know, It looks strange. Mathematically, this is the correct method to find the domain but how would I write that line then?
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  7. #7
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by Air View Post
    I know, It looks strange. Mathematically, this is the correct method to find the domain but how would I write that line then?
    x^2\leq 25 \implies \sqrt{x^2}\leq 5 and as \sqrt{x^2}=|x| we have |x|\leq 5 which is the same as -5\leq x \leq 5.
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