Editing Graph of a function (section)

== Examples ==

=== Functions of one variable ===

[[File:Three-dimensional graph.png|right|thumb|250px|Graph of the [[Function (mathematics)|function]] <math>f(x, y) = \sin\left(x^2\right) \cdot \cos\left(y^2\right).</math>]]

The graph of the function <math>f : \{1,2,3\} \to \{a,b,c,d\}</math> defined by
<math display=block>f(x)=
        \begin{cases}
              a, & \text{if }x=1, \\ d, & \text{if }x=2, \\ c, & \text{if }x=3, 
        \end{cases}
  </math>
is the subset of the set <math>\{1,2,3\} \times \{a,b,c,d\}</math>
<math display=block>G(f) = \{ (1,a), (2,d), (3,c) \}.</math>

From the graph, the domain <math>\{1,2,3\}</math> is recovered as the set of first component of each pair in the graph <math>\{1,2,3\} = \{x :\ \exists y,\text{ such that }(x,y) \in G(f)\}</math>.
Similarly, the [[Range of a function|range]] can be recovered as <math>\{a,c,d\} = \{y : \exists x,\text{ such that }(x,y)\in G(f)\}</math>.
The codomain <math>\{a,b,c,d\}</math>, however, cannot be determined from the graph alone.

The graph of the cubic polynomial on the [[real line]]
<math display=block>f(x) = x^3 - 9x</math>
is
<math display=block>\{ (x, x^3 - 9x) : x \text{ is a real number} \}.</math>

If this set is plotted on a [[Cartesian plane]], the result is a curve (see figure).
{{clear}}

=== Functions of two variables ===

[[File:F(x,y)=−((cosx)^2 + (cosy)^2)^2.PNG|class=skin-invert-image|thumb|250px|Plot of the graph of <math>f(x, y) = - \left(\cos\left(x^2\right) + \cos\left(y^2\right)\right)^2,</math> also showing its gradient projected on the bottom plane.]]

The graph of the [[trigonometric function]]
<math display=block>f(x,y) = \sin(x^2)\cos(y^2)</math>
is
<math display=block>\{ (x, y, \sin(x^2) \cos(y^2)) : x \text{ and } y \text{ are real numbers} \}.</math>

If this set is plotted on a [[Cartesian coordinate system#Cartesian coordinates in three dimensions|three dimensional Cartesian coordinate system]], the result is a surface (see figure).

Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane.  The second figure shows such a drawing of the graph of the function:
<math display=block>f(x, y) = -(\cos(x^2) + \cos(y^2))^2.</math>