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:

Square Root Function

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.

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) = $\sqrt{x}$ + 4
g(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.