# Function (in Algebra) A function in algebra is defined as **a relation between an input set, which is called the *domain*, and an output set, which is called the *codomain* (or the *range*)**, such that each **member of the domain is mapped to exactly one member of the codomain**. **domain -> range (a.k.a codomain)** If it so happens that exactly one member of the *codomain* is mapped to exactly one member of the *domain* (i.e. it's a 1-to-1 function), that function's [[Inverse (in Algebra)|inverse]] also exists. Functions can be further classified as: [[Polynomials]] (of which first degree polynomials are [[Linear Equations]], and second degree are [[Quadratics]]), [[Piecewise Functions]], [[Rational Functions]], [[Trigonometric Functions]], [[Periodic Functions]], and [[Exponential Functions]]. ^raszmz The above is the standard classification. However functions can be classified based on their [[Function Discontinuities|continuity]], [[Smooth Functions|smoothness]], and [[Graph Symmetry|symmetry]]. Each function can be transformed by composing it with other functions. For details see [[Function Transformations]]. The length of a section of a graph formed by a function can be determined using calculus. See [[Determining the Length of a Section of a Curve]]. ## Critical Points A function's *critical point* is any point where it's [[Derivative]] is 0. Those points are candidates for local (and global) minimums and maximums. ^g14unt When a critical point is also a function's inflection point (where second derivative is 0, (see [[#Concavity]]), it is known as a *saddle point*. [[The Second Derivative Test]] can be used to determine if a particular critical point is a local maximum or minimum. ## Concavity A section of a graph of a function can be described as being concave up (shaped like the letter U) or concave down. The second derivative can be used to determine a function's concavity at a particular point. If $f''(x) > 0$, the function is concave up at that point, if $f''(x) < 0$ then the function is concave down. If $f''(x)=0$ then that point is called the function's *inflection point* – a point where concavity changes. ^xi1i4g ## Horizontal and Vertical Line Tests Functions with real coefficients can be graphed using a 2D real number plane (the Cartesian Plane). One popular way of testing if a graph can be expressed as a function is the so-called *vertical line test*. **If any vertical line intersects the graph only once (or not at all), then the graph represents a function.** Similarly, the *horizontal line test* can be used to test if a function has an [[Inverse (in Algebra)|inverse]]. **If any horizontal line intersects the graph only once (or not at all), then the function has an inverse.**