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Math Help - Graphing square root functions

  1. #1
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    Graphing square root functions

    Hi!

    I can tell the domain from the graph of these functions but how do I tell the range from their graphs?
    h(x) = -\sqrt{x + 2} and f(x) = \sqrt{2 - x} - 1

    I can find the domain easily: h(x) = x + 2 \ge 0, x \ge -2


    Also, why does the graph of h(x) = -\sqrt{x + 2} flip over the x-axis and the graph of f(x) = \sqrt{2 - x} - 1 flip over the y-axis?
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  2. #2
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    Re: Graphing square root functions

    Think about the transformations to a function f(x). -f(x) is a reflection over the x-axis while f(-x) is a reflection over the y-axis.

    As for working out the range, for a standard function \displaystyle \begin{align*} f(x) = \sqrt{x} \end{align*}, the range is \displaystyle \begin{align*} f \in [0, \infty) \end{align*}. Why? Now what transformations have been applied and how does this affect the range?
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    Re: Graphing square root functions

    Quote Originally Posted by Prove It View Post
    As for working out the range, for a standard function \displaystyle \begin{align*} f(x) = \sqrt{x} \end{align*}, the range is \displaystyle \begin{align*} f \in [0, \infty) \end{align*}. Why? Now what transformations have been applied and how does this affect the range?
    Since the range is dependent on the domain...
    The range of the square root function lies in f \in [0, \infty) because we are finding the principal root of the non-negative value x.

    Transformation applied: square root function.
    Effect: Take only principal root.
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  4. #4
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    Re: Graphing square root functions

    No, I meant you start with the square root function and then apply some transformations to get the functions you have been given. What are these transformations?
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    Re: Graphing square root functions

    Quote Originally Posted by Prove It View Post
    No, I meant you start with the square root function and then apply some transformations to get the functions you have been given. What are these transformations?
    For h(x) = -\sqrt{x + 2}, means h(x) = -h(x).

    Begin: \sqrt{x}
    + 2: Shifts left, (-2,0)
    - : Flips over x-axis, point (-2,0) remains the same.

    For f(x) = \sqrt{2 - x} - 1. \Rightarrow \sqrt{-(x - 2)} - 1
    Begin: \sqrt{x}
    - 2: Shifts right, (2,0)
    - inside radical : Flips over y-axis, (2,0)
    - 1 : Pushes graph downward, (2,-1)
    Last edited by Unreal; April 28th 2013 at 06:06 PM.
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  6. #6
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    Re: Graphing square root functions

    Correct, so now what are the ranges of these functions?
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    Re: Graphing square root functions

    Range refers to y-values.

    h(x) = -\sqrt{x + 2}. \Rightarrow \text{Point}\ (2,0)

    Range: \[0,\infty)

    f(x) = \sqrt{2 - x} - 1. \Rightarrow \text{Point}\ (2, -1)

    Range: \[-1, \infty)
    Last edited by Unreal; April 28th 2013 at 05:55 PM.
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  8. #8
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    Re: Graphing square root functions

    I agree with your range for f(x) but not your range for h(x).
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    Re: Graphing square root functions

    Quote Originally Posted by Prove It View Post
    I agree with your range for f(x) but not your range for h(x).
    h(x) flips over the x-axis, so the y-values stop at 0.
    Range: \(-\infty,0]

    The range of h(x) seems less intuitive, maybe due to the graph flipping...your thoughts?
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  10. #10
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    Re: Graphing square root functions

    This is correct. And I don't think it's counter-intuitive at all actually, it should be obvious that when there's a reflection in one of the axes then either the domain or range will be inverted.
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