Root function

Functions: Power functions and root functions

Root function

The simplest root function is the function \[f(x)=\sqrt{x}\]

The table with this root function is (all roots rounded to 2 decimals):

#\begin{array}{c|c|c|c|c|c|c|c}
x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline y & 0 & 1 & 1.41 & 1.73 & 2 & 2.24 & 2.45
\end{array}#

The graph of the function is half a parabola with origin #\rv{0,0}#.

Since the root is only defined for non-negative numbers, the domain of the root function is equal to the interval #\ivco{0}{\infty}#.

Since the root of a non-negative number is a non-negative number in itself, the range is also equal to the interval #\ivco{0}{\infty}#.

Root as powerWe have seen previously that #\sqrt{x}=x^{\frac{1}{2}}#. Hence, we can also write the root function as \[f(x)=x^{\frac{1}{2}}\]

With this #f(x)# is a power function. A power function is a function of the form #y=\blue a x^{\orange{n}}# and in this case we have #\blue a=1# and #\orange n =\tfrac{1}{2}#.

Take a look at the function #f(x)=\sqrt{x}#. Does the point #\rv{5, 2.24}# lie on the graph on this function?
In here, you can round the #y#-value of the point to #2# decimals, if needed.

Yes

We substitute #x=5# in the formula. This is done in the following way:
\[f(5)=\sqrt{5}=2.24\]
Hence, #\rv{5, 2.24}# is a point on the graph.

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