Graph of the Function $y = x^x$

$y = x^x$ has the same variable in the base and in the exponent. It is neither a power function like $x^{a}$ nor an exponential like $a^{x}$, and its true nature only appears once it is rewritten as $x^x = e^{x \ln x}$.

Domain

Over the reals the domain is restricted to $x > 0$. For negative $x$ the logarithm in $e^{x \ln x}$ is undefined, and $(-0.5)^{-0.5}$, for instance, is not a real number. Negative integers do give values, such as $(-1)^{-1} = -1$, but they form no interval and so cannot be drawn as a curve.

Behavior at the origin

As $x \to 0^{+}$ we have $x \ln x \to 0$, so the limit is as follows.

The indeterminate form $0^0$ settling at $1$ is what makes the left end of the graph rise toward height $1$. Since $x = 0$ is not in the domain, that point is a hole.

Monotonicity and extrema

The derivative is $y' = x^x(\ln x + 1)$. As $x^x > 0$, the sign comes from $\ln x + 1$, giving a minimum, both local and global, at $x = \dfrac{1}{e}$. Its value is as follows.

The curve falls from $1$ at the left end down to $0.692$ before turning upward, a shape that surprises most people on first sight.

Concavity

The second derivative is $y'' = x^x\left[ (\ln x + 1)^2 + \dfrac{1}{x} \right]$. The bracket is positive for every $x > 0$, so the curve is concave up throughout and has no inflection point.

Rate of growth

At the integers the values are simply $n^n$. This outgrows every exponential $a^{x}$, because the base itself keeps increasing. It is also closely tied to the factorial: Stirling's formula $n! \approx \sqrt{2\pi n}\left( \dfrac{n}{e} \right)^{n}$ shows that $n!$ grows at essentially the rate of $\dfrac{n^n}{e^n}$.

The sophomore's dream

The definite integral from $0$ to $1$ has a strikingly simple series representation.

Similarly $\int_0^1 x^{-x}\,dx = \sum_{n=1}^{\infty} \dfrac{1}{n^{n}} \approx 1.2913$. Because the integrals turn straight into series of the same shape, these identities are known as the sophomore's dream. They follow from expanding the integrand $e^{\pm x \ln x}$ as an exponential series and integrating term by term.

Relation to other functions

The exponent $x \ln x$ is worth studying in its own right, and its minimum point is exactly the minimum point of $x^x$. Exchanging base and exponent gives $x^{1/x} = e^{(\ln x)/x}$, whose maximum is $e^{1/e} \approx 1.445$ at $x = e$.

The pairing of the minimum at $\dfrac{1}{e}$ with the maximum at $e$ is a pleasant symmetry.