Graph of the Fourth Root Function $y = \sqrt[4]{x}$

$y = \sqrt[4]{x}$ is the non-negative number whose fourth power is $x$, also written $x^{1/4}$¹. It can be seen as taking the square root twice: $\sqrt[4]{x} = \sqrt{\sqrt{x}}$.

Domain and range

• The domain is $x \geq 0$
• The range is $y \geq 0$
• The function increases monotonically
• The curve is concave down

Because $4$ is even, negative numbers have no real fourth root. Larger inputs give larger outputs, though the rise becomes ever gentler.

Monotonicity and the origin

The derivative $y' = \dfrac{1}{4}x^{-3/4}$ tends to $+\infty$ as $x \to 0^{+}$, so the tangent at the origin is vertical and the curve then leans over to the right. The second derivative is negative, so the graph is concave down throughout.

Concavity here means that each further increase in input buys less and less: raising $x$ from $16$ to $81$ lifts $y$ only from $2$ to $3$.

Notable points

Each time $x$ is multiplied by $16$, $y$ doubles.

Compared with the square root

The smaller the exponent $\dfrac{1}{n}$, the sharper the rise near the origin and the flatter the curve far away.

Inverse function

It is the inverse of $y = x^4$ restricted to $x \geq 0$, the reflection of that curve across the line $y = x$, and a member of the family $x^{1/n}$.

Applications

• The Stefan-Boltzmann law makes temperature proportional to the fourth root of radiated energy²
• The fourth-root transform, used in statistics to stabilise the spread of a distribution

1. Nth root, Wikipedia
2. Stefan-Boltzmann law, Wikipedia