There are many different types of functions in mathematics, some of the most common include:

- Linear functions: These functions have the form y = mx + b, where m is the slope of the line and b is the y-intercept. Linear functions have a constant rate of change, and their graph is a straight line.
- Quadratic functions: These functions have the form y = ax^2 + bx + c, where a, b and c are constants. Quadratic functions have a parabolic shape and their vertex is at the origin. They are also called parabolic functions
- Polynomial functions: These functions have the form y = a_nx^n + a_{n-1}x^{n-1} + … + a_1x + a_0, where n is the degree of the polynomial, and the a_i are constants. The graph of a polynomial function is a smooth curve that can be concave up or concave down.
- Exponential functions: These functions have the form y = a^x, where a is a constant greater than 0 and not equal to 1. The graph of an exponential function is a smooth curve that increases at an increasing rate.
- Logarithmic functions: These functions have the form y = log_a(x) , where a is a constant greater than 0 and not equal to 1. The graph of a logarithmic function is a smooth curve that decreases at a decreasing rate.
- Trigonometric functions: These functions have the form y = f(x) = sin(x), cos(x), tan(x) etc. Trigonometric functions are periodic, oscillating between -1 and 1. They are also used in physics, engineering and other fields.
- Rational functions: These functions have the form y = f(x) = (P(x))/(Q(x)) where P(x) and Q(x) are polynomials. The graph of a rational function can have asymptotes, holes and vertical lines.
- Piecewise functions: These functions are defined by multiple different rules for different ranges of the input. They are defined by multiple equations for multiple different ranges of x.

These are some of the most common types of functions in mathematics. Each type of function has its own unique properties and behaviors, and they are all used in different areas of mathematics and other fields.