page_title: Inverse function
section_title: Graph of the inverse
hierarchical_section_title: Inverse function / Inverses in calculus / Graph of the inverse
caption_attribution_description: The graph of a function y = ƒ(x) and its inverse.
context_page_description: In mathematics, an inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g = x.
  As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure, to reverse this and get x back from some output value, say y, we should undo each step in reverse order. In this case it means we should add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by,
  With y = 5x − 7 we have that f(x) = y and g = x.
  Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse, it must have the property that for every y in Y there is one, and only one x in X so that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f.
context_section_description: If  f is invertible, then the graph of the function
  is the same as the graph of the equation
  This is identical to the equation y = f(x) that defines the graph of f, except that the roles of x and y have been reversed. Thus the graph of f ⁻¹ can be obtained from the graph of f by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line
  y = x.
caption: The graphs of y = f(x) and y = f −1(x). The dotted line is y = x.
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