# Graphing square root functions

• Apr 28th 2013, 04:51 PM
Unreal
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?
$\displaystyle h(x) = -\sqrt{x + 2}$ and $\displaystyle f(x) = \sqrt{2 - x} - 1$

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

Also, why does the graph of $\displaystyle h(x) = -\sqrt{x + 2}$ flip over the x-axis and the graph of$\displaystyle f(x) = \sqrt{2 - x} - 1$ flip over the y-axis?
• Apr 28th 2013, 05:01 PM
Prove It
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 \displaystyle \begin{align*} f(x) = \sqrt{x} \end{align*}, the range is \displaystyle \displaystyle \begin{align*} f \in [0, \infty) \end{align*}. Why? Now what transformations have been applied and how does this affect the range?
• Apr 28th 2013, 05:14 PM
Unreal
Re: Graphing square root functions
Quote:

Originally Posted by Prove It
As for working out the range, for a standard function \displaystyle \displaystyle \begin{align*} f(x) = \sqrt{x} \end{align*}, the range is \displaystyle \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 $\displaystyle f \in [0, \infty)$ because we are finding the principal root of the non-negative value $\displaystyle x$.

Transformation applied: square root function.
Effect: Take only principal root.
• Apr 28th 2013, 05:23 PM
Prove It
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?
• Apr 28th 2013, 05:39 PM
Unreal
Re: Graphing square root functions
Quote:

Originally Posted by Prove It
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 $\displaystyle h(x) = -\sqrt{x + 2}$, means h(x) = -h(x).

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

For $\displaystyle f(x) = \sqrt{2 - x} - 1$. $\displaystyle \Rightarrow \sqrt{-(x - 2)} - 1$
Begin: $\displaystyle \sqrt{x}$
$\displaystyle - 2$: Shifts right, (2,0)
$\displaystyle -$ inside radical : Flips over y-axis, (2,0)
$\displaystyle - 1$ : Pushes graph downward, (2,-1)
• Apr 28th 2013, 05:42 PM
Prove It
Re: Graphing square root functions
Correct, so now what are the ranges of these functions?
• Apr 28th 2013, 05:52 PM
Unreal
Re: Graphing square root functions
Range refers to y-values.

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

Range: $\displaystyle \[0,\infty)$

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

Range: $\displaystyle \[-1, \infty)$
• Apr 28th 2013, 05:57 PM
Prove It
Re: Graphing square root functions
I agree with your range for f(x) but not your range for h(x).
• Apr 28th 2013, 06:11 PM
Unreal
Re: Graphing square root functions
Quote:

Originally Posted by Prove It
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: $\displaystyle \(-\infty,0]$

The range of h(x) seems less intuitive, maybe due to the graph flipping...your thoughts?
• Apr 28th 2013, 06:20 PM
Prove It
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.