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Power Function — Definition, Formula & Examples

A power function is a function of the form f(x)=kxnf(x) = kx^n, where kk is a nonzero constant and nn is a real number. Common examples include f(x)=x2f(x) = x^2, f(x)=x3f(x) = x^3, and f(x)=x=x1/2f(x) = \sqrt{x} = x^{1/2}.

A power function is any function that can be expressed as f(x)=kxnf(x) = kx^n for constants k≠0k \neq 0 and n∈Rn \in \mathbb{R}. The exponent nn determines the function's end behavior, symmetry, and domain. When nn is a positive integer, the function is a single-term polynomial (monomial).

Key Formula

f(x)=kxnf(x) = kx^n
Where:
  • kk = A nonzero real constant that scales the output
  • xx = The independent variable (input)
  • nn = A real number exponent that determines the function's shape

How It Works

The exponent nn controls the shape of a power function. If nn is a positive even integer, the graph is U-shaped and symmetric about the yy-axis. If nn is a positive odd integer, the graph passes through the origin with opposite behavior on each side. Negative exponents like n=−1n = -1 produce reciprocal-type curves with vertical and horizontal asymptotes. Fractional exponents like n=1/2n = 1/2 yield root functions whose domains are restricted to x≥0x \geq 0. The coefficient kk stretches or compresses the graph vertically and reflects it across the xx-axis when k<0k < 0.

Worked Example

Problem: Given the power function f(x)=3x4f(x) = 3x^4, evaluate f(2)f(2) and describe the function's symmetry.
Substitute: Replace xx with 2 in the formula.
f(2)=3(2)4f(2) = 3(2)^4
Compute the power: Calculate 24=162^4 = 16.
f(2)=3⋅16=48f(2) = 3 \cdot 16 = 48
Determine symmetry: Since the exponent n=4n = 4 is an even integer, f(−x)=3(−x)4=3x4=f(x)f(-x) = 3(-x)^4 = 3x^4 = f(x). The function is even, so its graph is symmetric about the yy-axis.
f(−x)=f(x)f(-x) = f(x)
Answer: f(2)=48f(2) = 48, and the function has even symmetry (symmetric about the yy-axis).

Why It Matters

Power functions form the building blocks of polynomials, so understanding them is essential in precalculus and calculus. They also model real-world relationships: area scales as x2x^2, volume as x3x^3, and gravitational force as x−2x^{-2}.

Common Mistakes

Mistake: Confusing power functions with exponential functions like f(x)=2xf(x) = 2^x.
Correction: In a power function, the variable xx is the base and the exponent is a constant (xnx^n). In an exponential function, the base is a constant and the variable is the exponent (axa^x). These have very different growth behaviors.