How to Find the Range of a Square Root Function

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Mathematical functions are written in terms of variables. A simple function y = f(x) contains an independent variable "x" (input) and a dependent variable "y" (output). The possible values for "x" are called the function's domain. The possible values for "y" are the function's range. A square root "y" of a number "x" is a number such as y^2 = x. This definition of the square root function imposes certain restrictions on the domain and range of the function, based on the fact that x cannot be negative

    Write down the complete square root function.

    For example: f(x) = y = SQRT( x^3 -8 )

    Set the input of the function to equal or greater than zero. From the definition y^2 = x; x must be positive, this is why you set the inequality to zero or greater than zero.Solve the inequality using algebraic methods. From the example:

    x^3 -8 >= 0 x^3 >= 8 x >= +2

    Since x must be greater or equal to +2, the domain of the function is [ +2, +infinite [

    Write down the domain. Replace values from the domain into the function to find the range. Start with the left boundary of the domain, and choose random points from it. Use these results to find a pattern for the range.

    Continuing the example: Domain : [ +2, +infinite [ at +2, y = f(x) = 0 at +3, y = f(x) = +19 ... at +10, y = f(x) = +992

    From this pattern, it is evident that as x goes up in value, f(x) also goes up. The dependent variable "y" grows starting at zero to "+infinite. This is the range.

    Range: [ 0, +infinite [


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