The inverse relationship for the square function y = x2 with its entire domain of all real numbers is y = $\pm$ $\sqrt{x}$ which is not a function. An inverse function can be defined by restricting the domain of the square function. Thus the square root function g (x) = $\sqrt{x}$ is the inverse for f(x) = x2 where x is a non negative number.

Same way restricting the domain as x $\leq$ 0, the negative of g(x), h(x) = - $\sqrt{x}$ becomes the inverse for the square function.

A square root function is an algebraic function as it is expressed as the square root of the variable. The parent square root function is given as
f(x) = $\sqrt{x}$
The graph of square root function is given below:

The characteristics of square root function can be stated as:

• The domain of the function is the set of all non negative real numbers.
• The range of the function is the set of all non negative real numbers.
• The graph meets both the x and y axis at (0, 0)
• The function is increasing or the graph is rising up in the entire domain of the function.

## Solving Square Root Functions

The transformed equations of square root function are given below detailing the transformation involved.

 Equation Transformation involved Example g(x) = a$\sqrt{x}$ Vertical stretch by a factor of a g(x) = 3$\sqrt{x}$ g(x) = $\sqrt{b}$x Vertical stretch by a factor of 1/b g(x) = $\sqrt{2}$x g(x) = -$\sqrt{x}$ Reflection over x - axis. Vertical reflection. g(x) = $\sqrt{-x}$ Reflection over y - axis.Horizontal reflection g(x) = $\sqrt{x}$ + k Vertical shift up by k units. When k is negative shift down by k units. g(x) = $\sqrt{x}$ + 4g(x) = $\sqrt{x}$ - 5 g(x) = $\sqrt{x-h}$ Horizontal shift to right by h units. g(x) = $\sqrt{x-3}$ g(x) = $\sqrt{x+h}$ Horizontal shift to left by h units. g(x) = $\sqrt{x+4}$

Examples for a composite transformation:

g(x) = $-3\sqrt{x-2}+5$
The transformations occur in the following order.
1. Reflection over x axis.
2. Vertical Stretch by factor 3
3. Horizontal shift to right by 2 units and vertical shift up by 5 units.

h(x) = -$\sqrt{2x +6}$ - 4
The equation has to be first rewritten as = -$\sqrt{2(x+3)}$ - 4. The transformations occur in the following order.

1. Reflection over x axis.
2. Horizontal stretch by factor $\frac{1}{2}$.
3. Horizontal shift to left by 3 units and vertical shift down by 4 units.

### Rational Function

 How to do Square Roots Rationalizing the Denominator with Square Roots Simplify Square Root Expressions Square Root Taylor Series Roots of a Quadratic Function A Square Root Law Even Functions Into Function What is a Function? Difference of Cube Roots Find Roots of Quadratic Equation
 3 Square Root Calculator Calculator Functions Calculate Inverse Function