Master every function type with visual explanations, key properties, domain & range, and interactive graphs. From linear to trigonometric — all in one place.
Foundation of algebra — constant, linear, quadratic, cubic and beyond.
f(x) = c
A constant function outputs the same value for every input. Its graph is a horizontal line.
f(x) = x
The identity function returns exactly its input. It's the simplest linear function.
f(x) = ax + b
A linear function has a constant rate of change. Its graph is a straight line.
f(x) = ax² + bx + c
A quadratic function creates a parabola. The vertex is the maximum or minimum point.
f(x) = ax³ + bx² + cx + d
Cubic functions have an S-shaped curve with one inflection point and up to two local extrema.
f(x) = Σaₙxⁿ
A degree-n polynomial has at most n roots and n-1 local extrema. End behavior is determined by the leading term.
Power and root functions — integer and fractional exponents.
f(x) = axⁿ
Power functions with integer exponents. Even powers are U-shaped; odd powers are S-shaped.
f(x) = ax^(p/q)
Fractional exponents represent roots. x^(1/2) = √x, x^(1/3) = ∛x, etc.
f(x) = √x
The square root function is the inverse of x². It only accepts non-negative inputs.
f(x) = ∛x
The cube root is defined for all real numbers, unlike the square root.
Rational functions — ratios of polynomials with asymptotes.
Exponential and logarithmic functions — inverse pair.
f(x) = a·bˣ
Exponential functions model growth (b > 1) or decay (0 < b < 1) at a constant percentage rate.
f(x) = eˣ
The natural exponential is its own derivative — the unique function where f'(x) = f(x).
f(x) = logₐ(x)
The logarithm is the inverse of the exponential. It answers: "to what power must we raise a to get x?"
f(x) = ln(x)
The natural logarithm is the inverse of eˣ. Its derivative is 1/x.
Trigonometric and inverse trig functions — periodic and bounded.
f(x) = sin(x)
The sine function oscillates between -1 and 1 with period 2π. It models circular motion and waves.
f(x) = cos(x)
The cosine function is the horizontal component of circular motion. It's an even function.
f(x) = tan(x)
Tangent = sin/cos. It has vertical asymptotes where cos(x) = 0 and period π.
f(x) = arcsin(x)
Arcsin is the inverse of sin, restricted to [-π/2, π/2]. It answers: "what angle has this sine?"
f(x) = arccos(x)
Arccos is the inverse of cos, restricted to [0, π]. It gives the angle in the upper half.
f(x) = arctan(x)
Arctan is the inverse of tan. It's defined for all real x and has horizontal asymptotes at ±π/2.
Piecewise and composite functions — combining simpler functions.
Open the interactive graphing tool to explore any function in real time.
Open Function Visualizer