DEFINITION AND FORM
A power function is a function of the form
f(x) = xⁿ,
where 'n' is any real number. The exponent 'n' determines the properties of the function: its domain, range, the course of its graph, as well as its symmetry and monotonicity. Power functions include natural numbers, as well as negative and rational numbers in the exponent.
Depending on the value of the exponent 'n', we distinguish several cases with characteristic shapes and behaviors of the function.
EXAMPLES BY EXPONENT
n ∈ ℕ (NATURAL NUMBERS)
- f(x) = x² – parabola, an even function, graph is symmetrical with respect to the y-axis.
- f(x) = x³ – an odd function, graph is symmetrical with respect to the origin.
n = 1
- f(x) = x – a linear function, the identity function.
n < 0 (NEGATIVE NUMBERS)
- f(x) = x⁻¹ = 1/x – a rational function with asymptotes (x ≠ 0), an odd function.
- f(x) = x⁻² = 1/x² – positive for all x ≠ 0, an even function.
n = 0
- f(x) = x⁰ = 1 for x ≠ 0 – a constant function (usually defined as f(x) = 1).
n ∈ ℚ (RATIONAL NUMBERS)
- f(x) = x¹ᐟ² = √x (square root of x) – defined only for x ≥ 0, increasing, neither even nor odd.
- f(x) = x¹ᐟ³ = ∛x (cube root of x) – defined on ℝ (all real numbers), an odd function.
PROPERTIES BASED ON THE EXPONENT
- If 'n' is an odd integer, the function is odd: f(–x) = –f(x) holds true.
- If 'n' is an even integer, the function is even: f(–x) = f(x) holds true.
- If n < 0, the function has asymptotes, often not defined for x = 0.
- If 'n' is rational (like a fraction m/k), the function is defined only where the k-th root is defined (e.g., √x only for x ≥ 0).
DOMAIN AND RANGE
These depend on the exponent:
- f(x) = xⁿ, n ∈ ℕ → Domain (Df) = ℝ, Range (Rf) = ℝ (if n is odd), Rf = [0, ∞) (if n is even and positive)
- f(x) = x⁻¹ → Df = ℝ \ {0}, Rf = ℝ \ {0} (real numbers excluding 0)
- f(x) = √x → Df = [0, ∞), Rf = [0, ∞)
EXAMPLE
Let f(x) = x³.
- Df = ℝ, Rf = ℝ.
- f(–2) = –8, f(2) = 8 → the function is odd.
- The graph passes through the origin and is increasing over its entire domain.
CONCLUSION
The power function is one of the basic families of functions. Its form and properties are directly dependent on the exponent. With them, we study different types of growth, symmetry, and the behavior of the function in its domains. Due to its simple structure and diversity depending on the values of 'n', this function holds an important place in mathematical analysis. analysis.
