Let’s graph the following function:
First we have to consider the domain of the function. We must note that we cannot
have a negative value under the square root sign or we will end up with a complex number. Therefore,
we set whatever is under the root sign great than or equal to 0.
Remember that when we divide by a negative number, we flip the inequality. This
result means that the domain of x, or the input, is any value less than or equal
Next, we can go ahead and plot our points, but we must be careful not to plot points
that are close together since we will not get an accurate picture. A method we can
use is to set the function equal to different positive integers to see what their
x value is.
For instance, we want to see what x value gives a y value of 4, we can ask ourselves,
“The square root of what value gives us 4?” We know it is 16. Then we can ask, “What
value when subtracted from 2 gives us 16?” We can see 2 – (-14) = 16. Therefore,
to get a y value of 4, we need an x value of -14.
We can see why x cannot be greater than 2 on the graph, and we can also see why
there are no negative y values. If x is greater than 2, we would end up with a complex
number and we cannot yield a negative y value from an expression under a square
We can also see that this looks somewhat like a sideways parabola, with the negative
y values ommited. This is true, and if we square both sides of the function and
isolate x, we end up with the equation of the parabola in terms of y.
This way, our range is not restricted to only the positive y values. However, we
must realize that this equation is different from original function, because it
is in fact not a function. Recall that to be a function, the image must pass the
vertical line test.
It is important to be aware of this difference, and understand how radical functions
in terms of x algebraically and geometrically relate to equations in terms of y.
Next, let’s graph the functions:
Remember, the first thing we need to do is see if we have any restrictions on our
domain. We cannot have a negative value inside of the square root, so we could set
both expressions inside the root sign greater or equal to 0. Instead, we can graph
both of the expressions inside the roots as functions on the graph and see if any
x values yield a negative y value.
We can see the graph of g(x) = x2+9 is always above the x axis,
meaning that for all x values, the functions yields positive y values. Since we
will always have a positive y value for this function, there are no restrictions
on our domain.
The graph of f(x) = x2-9 dips below the x axis in between -3 and
3. This means we have restrictions on our domain in between the values of x = -3
and x = 3.
When dealing with
polynomials inside a root sign, graphing the polynomial function is the
easiest way to see where the function dips below the x axis and find the x intercepts.
Wherever the graph is negative on the x axis is the restriction on the domain of
the original function.
To graph the functions, we need to keep the domain in mind for f(x) and graph
points less than -3 and greater than 3. For the the graph of g(x), we can
plot any x value we want. We can do this by making an xy table or plugging the equations
into a graphing calculator.
If we observe the nature of the graphs, they look very similar to two different
without their negative y values. We can do some manipulation and see that the functions
can be represented as hyperbolic equations.
Every radical function will be part of a
conic section. This is because when we manipulate the function to be in
terms of both x and y, we will always have a y2. Let’s do another
example illustrating this point.
Next, graph the function of:
First, we check the domain by graphing the expression inside the root and setting
it equal to y.
We can see that the x intercepts are (-4,0) and (4,0), and when x is
less than -4 and greater than positive 4, we have negative y values.
These are our restrictions.
Let’s plot our original function with our domain restrictions in mind (in other
words, plot points between x = -4 and x = 4).
We should observe that this is a semicircle missing it’s negative y values. With
some manipulation, we can come up with the equation of a circle with radius 4.
We have evaluated radical functions involving square roots. When graphing these
functions, we must be aware of the domain before we graph them. Some radical functions,
however, will never have domain constraints. Let’s look at a cube-root function.
By way of example, graph the cube-root function:
There are no domain restraints because we can take the cube root of a negative
number. Therefore, our domain is “all real numbers,” and we can plot any x value
What if we have a function with a 4th root such as
We cannot have a negative y value for any input. For example, (2)4 and
(-2) 4 both yield positive 16. If we plug in -16 for x, we will get a
complex number. This means we need to think about our domain before we graph.
We can deduce that for any radical function
If n is odd, our domain is not restricted. If n is even,
we must consider constraints on our domain.
For the next example, we want to find the domain of the function:
We can graph the inside functions, but let’s set the expression inside the radical
to greater or equal to 0.
We can then right the domain as [-∞,-3] U [3,∞] to indicate the
domain is any x value less than -3 and greater than 3.
Sometimes the domain of a radical function will not have any positive y values,
and therefore the graph will not exist for real numbers.
For example, find the domain and solution set to the following function
We can already see by inspection that the expression inside the square root will
never be positive. Let’t set it greater or equal to 0.
We cannot have a square root of a negative, so the domain is undefined and therefore
the image of the function is undefined on the real xy plane. We can also check by
graphing the expression and setting it equal to y.
Finding Zeros of Radical Functions
Finding the zeros is another way of saying finding the roots. Finding the zeros
of radical functions is unique because sometimes the roots that we find do not actually
satisfy the function. These roots are called extraneous zeros.
The strategy for finding roots of radical functions is to isolate the radical expression
and then square both sides to solve for x. In doing this, we square a quantity which
will get us the same result if we square its opposite, which does not satify the
original function. Like we observed before in the first three examples, the equation
we ended up with when we solved for both x and y is different from the original
function we had. Because of this, we must always check our results when finding
Last, let’s look at the function:
First, to find our domain, we set each expression under the radical greater or equal
Since we have two constraints, we take the one that is most restrictive, and thus
the domain is [-7/3, ∞].
To find our x intercepts, we set the function equal to 0 and solve for x.
We have two roots – x = 3 and x = -2. Lets plug them in and check
to see if they satisfy the function.
Since our function equals 0, 3 is a root.
Since our function does not equal 0, -2 is not a root.
Looking at our function, we can clearly see our x intercept and our domain restriction.