Graph of a function

<h2 id="definition">Definition</h2>
<p>Given a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 from a set X (the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>) to a set Y (the <a href="/facts/Codomain/MGc43nwQ">codomain</a>), the graph of the function is the set<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>

G
        (
        f
        )
        =
        {
        (
        x
        ,
        f
        (
        x
        )
        )
        :
        x
        ∈
        X
        }
        ,
      
    
    {\displaystyle G(f)=\{(x,f(x)):x\in X\},}

which is a subset of the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> 
  
    
      
        X
        ×
        Y
      
    
    {\displaystyle X\times Y}
  
. In the definition of a function in terms of <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
</p>
<h2 id="examples">Examples</h2>
<h3>Functions of one variable</h3>

<p>The graph of the function 
  
    
      
        f
        :
        {
        1
        ,
        2
        ,
        3
        }
        →
        {
        a
        ,
        b
        ,
        c
        ,
        d
        }
      
    
    {\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}}
  
 defined by

f
        (
        x
        )
        =
        
          
            {
            
              
                
                  a
                  ,
                
                
                  
                    if 
                  
                  x
                  =
                  1
                  ,
                
              
              
                
                  d
                  ,
                
                
                  
                    if 
                  
                  x
                  =
                  2
                  ,
                
              
              
                
                  c
                  ,
                
                
                  
                    if 
                  
                  x
                  =
                  3
                  ,
                
              
            
            
          
        
      
    
    {\displaystyle f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}}

is the subset of the set 
  
    
      
        {
        1
        ,
        2
        ,
        3
        }
        ×
        {
        a
        ,
        b
        ,
        c
        ,
        d
        }
      
    
    {\displaystyle \{1,2,3\}\times \{a,b,c,d\}}

G
        (
        f
        )
        =
        {
        (
        1
        ,
        a
        )
        ,
        (
        2
        ,
        d
        )
        ,
        (
        3
        ,
        c
        )
        }
        .
      
    
    {\displaystyle G(f)=\{(1,a),(2,d),(3,c)\}.}

</p><p>From the graph, the domain 
  
    
      
        {
        1
        ,
        2
        ,
        3
        }
      
    
    {\displaystyle \{1,2,3\}}
  
 is recovered as the set of first component of each pair in the graph 
  
    
      
        {
        1
        ,
        2
        ,
        3
        }
        =
        {
        x
        :
         
        ∃
        y
        ,
        
           such that 
        
        (
        x
        ,
        y
        )
        ∈
        G
        (
        f
        )
        }
      
    
    {\displaystyle \{1,2,3\}=\{x:\ \exists y,{\text{ such that }}(x,y)\in G(f)\}}
  
.
Similarly, the <a href="/facts/Range_of_a_function/66CeTV6u">range</a> can be recovered as 
  
    
      
        {
        a
        ,
        c
        ,
        d
        }
        =
        {
        y
        :
        ∃
        x
        ,
        
           such that 
        
        (
        x
        ,
        y
        )
        ∈
        G
        (
        f
        )
        }
      
    
    {\displaystyle \{a,c,d\}=\{y:\exists x,{\text{ such that }}(x,y)\in G(f)\}}
  
.
The codomain 
  
    
      
        {
        a
        ,
        b
        ,
        c
        ,
        d
        }
      
    
    {\displaystyle \{a,b,c,d\}}
  
, however, cannot be determined from the graph alone.
</p><p>The graph of the cubic polynomial on the <a href="/facts/Real_line/6eq42xX5">real line</a>

f
        (
        x
        )
        =
        
          x
          
            3
          
        
        −
        9
        x
      
    
    {\displaystyle f(x)=x^{3}-9x}

{
        (
        x
        ,
        
          x
          
            3
          
        
        −
        9
        x
        )
        :
        x
        
           is a real number
        
        }
        .
      
    
    {\displaystyle \{(x,x^{3}-9x):x{\text{ is a real number}}\}.}

</p><p>If this set is plotted on a <a href="/facts/Cartesian_plane/lkTqvMmB">Cartesian plane</a>, the result is a curve (see figure).
</p>

<h3>Functions of two variables</h3>

<p>The graph of the <a href="/facts/Trigonometric_function/czej7ur0">trigonometric function</a>

f
        (
        x
        ,
        y
        )
        =
        sin
        ⁡
        (
        
          x
          
            2
          
        
        )
        cos
        ⁡
        (
        
          y
          
            2
          
        
        )
      
    
    {\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})}

{
        (
        x
        ,
        y
        ,
        sin
        ⁡
        (
        
          x
          
            2
          
        
        )
        cos
        ⁡
        (
        
          y
          
            2
          
        
        )
        )
        :
        x
        
           and 
        
        y
        
           are real numbers
        
        }
        .
      
    
    {\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ and }}y{\text{ are real numbers}}\}.}

</p><p>If this set is plotted on a <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">three dimensional Cartesian coordinate system</a>, the result is a surface (see figure).
</p><p>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:

f
        (
        x
        ,
        y
        )
        =
        −
        (
        cos
        ⁡
        (
        
          x
          
            2
          
        
        )
        +
        cos
        ⁡
        (
        
          y
          
            2
          
        
        )
        
          )
          
            2
          
        
        .
      
    
    {\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}.}

</p>
<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Asymptote/nvLPuxKa">Asymptote</a></li>
<li><a href="/facts/Chart/2xshvBX0">Chart</a></li>
<li><a href="/facts/Plot_(graphics)/dxuSqKN9">Plot</a></li>
<li><a href="/facts/Concave_function/HxVEBkce">Concave function</a></li>
<li><a href="/facts/Convex_function/IbzG5SLF">Convex function</a></li>
<li><a href="/facts/Contour_line/35Qten5r">Contour plot</a></li>
<li><a href="/facts/Critical_point_(mathematics)/MrEk9PnY">Critical point</a></li>
<li><a href="/facts/Derivative/7Ito70Dh">Derivative</a></li>
<li><a href="/facts/Epigraph_(mathematics)/qGYcHa8t">Epigraph</a></li>
<li><a href="/facts/Normal_(geometry)/jhtLIhcu">Normal to a graph</a></li>
<li><a href="/facts/Slope/a2L3k0j5">Slope</a></li>
<li><a href="/facts/Stationary_point/f56GSgBj">Stationary point</a></li>
<li><a href="/facts/Tetraview/HSPGLruh">Tetraview</a></li>
<li><a href="/facts/Vertical_translation/KlpWmnfX">Vertical translation</a></li>
<li><a href="/facts/Y-intercept/oJvV25PI">y-intercept</a></li></ul>

<h2 id="further-reading">Further reading</h2>

<ul><li>Zălinescu, Constantin (30 July 2002). <a href="https://archive.org/details/convexanalysisge00zali_934"><i>Convex Analysis in General Vector Spaces</i></a>. River Edge, N.J. London: <a href="/facts/World_Scientific_Publishing/qJPTP0Yk">World Scientific Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4488-15-0. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1921556">1921556</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/285163112">285163112</a> – via <a href="/facts/Internet_Archive/PHStU8Lo">Internet Archive</a>.</li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Function plots.

<ul><li>Weisstein, Eric W. "<a href="http://mathworld.wolfram.com/FunctionGraph.html">Function Graph</a>." From MathWorld—A Wolfram Web Resource.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Charles C Pinter (2014) [1971]. A Book of Set Theory. Dover Publications. p. 49. ISBN 978-0-486-79549-2. <a href="978-0-486-79549-2" target="_blank">978-0-486-79549-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>T. M. Apostol (1981). Mathematical Analysis. Addison-Wesley. p. 35. <a href="/wiki/Tom_M._Apostol" target="_blank">/wiki/Tom_M._Apostol</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>P. R. Halmos (1982). A Hilbert Space Problem Book. Springer-Verlag. p. 31. ISBN 0-387-90685-1. <a href="0-387-90685-1" target="_blank">0-387-90685-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>D. S. Bridges (1991). Foundations of Real and Abstract Analysis. Springer. p. 285. ISBN 0-387-98239-6. <a href="0-387-98239-6" target="_blank">0-387-98239-6</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Graph of a function open-in-new

Graph of a function