Square Root (Radical) Functions

The Square Root Function

The simplest radical function is $f(x)=\sqrt{x}$. It takes an input $x$ and returns its principal (nonnegative) square root. Here are some outputs.

Plot those points and connect them. You get half of a parabola lying on its side.

Notice three things. The graph starts at $(0,0)$ and never goes to the left of the $y$-axis. It never goes below the $x$-axis. It keeps rising, but more and more slowly. So the domain is $x\ge 0$ and the range is $y\ge 0$.

To evaluate a radical function, substitute the input into the radicand and take the root.

Finding the Domain

So finding the domain is always the same two steps. Write "radicand $\ge 0$," then solve the inequality.

Look at the graph. To the left of $x=8$ there is no curve at all. The function is not "zero" there; it is undefined.

When the Radicand Is Quadratic

What if the radicand is something like $x^2-9$? Then the domain condition is a quadratic inequality, and the answer can be an interval between two numbers or two pieces outside them.

Here is the method.

1. Write radicand $\ge 0$.

2. Find where the radicand equals $0$ (factor it). These are the boundary points.

3. Picture the parabola $y=g(x)$. You want the part that is on or above the $x$-axis.

4. Check with a test point.

Now a parabola that opens down. Here is $g(x)=-x^2+4x+5=-(x-5)(x+1)$. It is nonnegative only between its zeros.

Shifting and Reflecting the Graph

Slide $h$ and $k$ and watch the starting point. A handy way to match a graph to an equation is to use the key points. From the start $(h,k)$, the curve passes through $(h+1,\,k+1)$, $(h+4,\,k+2)$ and $(h+9,\,k+3)$. For example, $\sqrt{x-2}$ starts at $(2,0)$, then goes through $(3,1)$ and $(6,2)$.

When the radicand has a coefficient on $x$, find the starting $x$-value by solving radicand $=0$. The range comes from the number added outside.

Solving for $x$ and Finding Intercepts

Evaluating gives an output from an input. Often you want to go the other way: given the output, find the input. For a radical function, you undo the square root by squaring.

1. Isolate the radical on one side.

2. Square both sides.

3. Solve the resulting equation.

4. Check your answer in the original equation.

Intercepts. The $y$-intercept is $f(0)$, but only if $0$ is in the domain. The $x$-intercept is found by solving $f(x)=0$.

Intersections and Unknown Parameters

To find where two graphs meet, set their expressions equal and solve. Then check the solution in the original equations. It must also be in the domain of both functions.

You can also use points on the graph to find unknown numbers in the equation.

Piecewise Radicals, Continuity, and Symmetry

A piecewise function uses a different rule on different parts of the domain. Radical functions are often combined this way. Here are three questions you may need to answer.

Domain: it is the union of the domains of the pieces, each restricted to its own interval.

Continuity at $x=c$: the graph can be drawn through $x=c$ without lifting your pencil. For a piecewise function, check that the pieces meet at the same point there.

Symmetry: a function is even if $f(-x)=f(x)$ for all $x$ in the domain (the graph is symmetric about the $y$-axis). It is odd if $f(-x)=-f(x)$ (the graph is symmetric about the origin). To test it, compute $f(-x)$ for $x>0$ and for $x<0$, using the correct piece each time.

Radical Functions in the Real World

A square root appears whenever one quantity depends on the square of another and you solve for the second. Think of falling objects, pendulums, waves, and the horizon distance.

Here is the pendulum formula $T(L)=2\pi\sqrt{L/10}$ ($T$ is in seconds and $L$ is the string length in feet). Move along the curve and notice how it flattens out.

To evaluate, substitute. To solve for the input, isolate the root, square, and check.

Scaling and the Domain in Context

Scaling. Because $\sqrt{m^2\cdot x}=m\sqrt{x}$, to multiply the output of a square root function by $m$, you have to multiply the input by $m^2$.

Domain in context. The formula may have a restricted domain, and the real situation may restrict it even more (for example, counts must be whole numbers).

Rate of Change and Geometry

Recall the average rate of change of $f$ from $x=a$ to $x=b$: $$\frac{f(b)-f(a)}{b-a}.$$ For a radical function it is not constant. The graph is curved, so the rate depends on the interval.

Geometry. A square root turns area into length. If a face of a cube has area $A$, then its edge is $a=\sqrt{A}$. Since $V=a^3$, we can write the volume in terms of $A$ as $$V=(\sqrt{A})^3=A\sqrt{A}.$$

Back to the bridge: for $t(h)=\sqrt{h/5}$, the condition $h/5\ge 0$ means $h\ge 0$, so the domain is $[0,\infty)$. That matches reality, since there are no negative heights. A ball dropped from a 45-ft bridge takes $t(45)=\sqrt{9}=3$ seconds to land.

Summary