What is a radical function? A radical function is a function that contains roots. Examples of radical functions are f(x) = √x, g(y) = 3√(2y + 3) and h(z) = √(3z - 5). One of the most common radical functions is the square root function. We will start with the most basic square root function, f(x) = √x. We are interested its graph and its domain and range. First, we'll find the domain of the function. Recall that only values for x ≥ 0 satisfy √x. If x < 0, we will not get a real number solution for √x.
To graph f(x) = √x, we'll set up a table of values.
For x = 1, f(1) = √1 = 1
x = 4, f(4) = √4 = 2
x = 9, f(9) = √9 = 3
x = 16, f(16) = √16 = 4
We would plot this on a rectangular coordinate system with the
horizontal axis being x, the vertical axis being f(x). Note that the
vertical axis on a rectangular coordinate system is generally the y
axis. Notice we used values for x that are perfect squares. They are
much easier to graph.
Note that we only used the principal square root for the values of f(x). The reason for this will be noted
after the graph. Recall that the square root has both positive and
negative values (ie, √4 = 2 and -2). But the reason we did not use the
negative values on the graph is because we are assuring that it is a
function. If we graph the negative values, the graph is a parabola which
opens to the right. This would not pass the vertical line test and
would not be a function. Therefore the graph of the function f(x) = √x
are only the values in the first quadrant.
Knowing the graph of the basic square root function, f(x) = √x, will
help us graph many other radical functions. If h > 0, then f(x) = √(x
+h) and f(x) = √(x - h) are the graphs of f(x) = √x moved to the left
and right h units, respectively. For example, the graph of f(x) = √(x +
1) has an x-intercept is (-1, 0). When x = 0, f(x) = 1. The domain is x ≥
-1 because x < -1 will yield a negative under the radical and the
square root of a negative is not a real number. You can plot several
other points that makes √(x + 1) a perfect square. This graph is the
graph of f(x) = √x moved 1 unit to the left. It is said to be translated
1 unit to the left. The graph of f(x) = √(x - 1), similarly, would be
the graph of f(x) = √x moved 1 unit to the right.
We learned how to graph radical functions that are translations left and
right of the square root function, f(x) = √x. Now we will learn how to
graph radical functions that are that translations up and down of the
square root function f(x) = √x. If k >0 , then f(x) = √x - k and f(x)
= √x + k are the graphs of f(x) moved down k units and up k units,
respectively. Therefore the graph of f(x) = √x + 3 is the same as the
graph of f(x) = √x except it is translated 3 units up. Similarly, the
graph of f(x) = √x - 3 is the same as the graph of f(x) = √x except it
is translated 3 units down.
A function can have multiple translations. For example, the graph of
f(x) = √(x - 2) + 1 is the same as the graph of f(x) = √x except it is
translated 2 units right and 1 unit up. You will notice when graphing
that there is vertical line through x = 2. The domain of the function is
x ≥ 2. This shows that there are no points on the graph to the left of 2
on the x axis. The graph is in the first quadrant where both x and f(x)
The cube root function is a radical function defined as f(x) = 3√x.
To graph the function we want to choose values for x that are a perfect
cube. For example 1 is a perfect cube because 1(1)(1) = 1 and 8 is a
perfect cube because (2)(2)(2) = 8.
To graph f(x) = 3√x, we'll set up a table of values.
Fox x = 1, f(1) = 3√(1) = 1
x = -1, f(-1) = 3√(-1) = -1
x = 8, f(8) = 3√(8) = 2
x = -8, f(-8) = 3√(-8) = -2
x = 27, f(27) = 3√(27) = 3
x = -27, f(-27) = 3√(-27) = -3
The function contains values in the first and third quadrants and no values in the second and fourth
quadrants. As with the square root function, there are translations with the cube root function. The graph of f(x) = 3√x - 3 is the same as the graph of this function is the same the graph of f(x) = 3√x except it is translated down 3 units.
This guide should help clear any confusion on the topic of radical functions and graphing radical functions.