Power Functions

Take the simple letter $x$ and ask what happens when you raise it to different powers. Square it and you get the gentle bowl of a parabola. Cube it and the curve learns to dip below the axis. Flip the power negative and the graph splits in two and races toward invisible walls. Turn the power into a fraction and you have square roots and cube roots. Every one of these is the same idea wearing a different number, and once you can read the number you can picture the graph before you draw a single point.

What a power function actually is

A power function is anything of the form $y = x^n$, where the variable sits on the bottom and the power $n$ is a fixed number. That power $n$ is the whole personality of the graph. It decides the shape, where the graph is allowed to live, and how it behaves near the edges.

There are three families worth knowing, and they all hide inside that one little $n$.

• Positive whole powers like $x^2$, $x^3$, $x^4$. These are smooth curves that pass through the origin.
• Negative powers like $x^{-1}$, $x^{-2}$. A negative power means “one over”, so $x^{-2} = \frac{1}{x^2}$. These graphs have a forbidden value where the bottom would be zero.
• Fractional powers like $x^{1/2}$, $x^{1/3}$. A fractional power is a root, so $x^{1/2} = \sqrt{x}$ and $x^{1/3} = \sqrt[3]{x}$.

Odd powers versus even powers

For the whole number powers, the single most useful question is whether $n$ is odd or even.

An even power such as $x^2$ or $x^4$ turns every input positive, because a negative times a negative is positive. So the graph sits entirely above the axis, looks like a bowl, and is a mirror image across the $y$ axis. Because the left and right sides match, the same height is reached twice, which makes an even power many to one, so it has no inverse function unless you chop the domain in half.

An odd power such as $x^3$ or $x^5$ keeps the sign of the input. Feed in a negative and you get a negative out. The graph flows from bottom left to top right through the origin, never repeats a height, and so it is one to one. That is why $y = x^3$ has a proper inverse function but $y = x^2$ does not.

The two traps in the domain

Here is where careful students pull ahead. The maximal domain is the full set of $x$ values you are allowed to feed in, and only two things can ban a value.

First, you can never divide by zero. So any negative power, which is secretly a fraction, must throw out the value that makes the bottom zero. For $x^{-2} = \frac{1}{x^2}$ the banned value is $x = 0$, giving a domain of $\mathbb{R} \setminus \{0\}$.

Second, you can never take an even root of a negative. A power of $\frac{1}{2}$ is a square root and a power of $\frac{1}{4}$ is a fourth root, and both reject negatives, so their domain starts at zero: $[0, \infty)$. But an odd root is relaxed about signs. A power of $\frac{1}{3}$ is a cube root, and since a negative cubed is negative, you are allowed to feed in negatives. So $x^{1/3}$ has domain all of $\mathbb{R}$, and $(-8)^{1/3} = -2$ is perfectly valid.

Differentiating a power function

The good news is that one rule handles every case, fractions and negatives included. Bring the power down to the front as a multiplier, then knock the power down by one.

So $\frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2}$, which you tidy into $\frac{1}{2\sqrt{x}}$. The mistake examiners see again and again is keeping the original power, or worse, adding one to it, which is antidifferentiation in disguise. Subtract one from the power, never add. Tidy your negative powers back into fractions and roots so your final answer is easy to read and easy to mark.

See these moves in action in the Worked Examples tab, then test yourself in Try It.

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