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Home < More Algebra < 4 Functions << 23 Inverse Functions
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Inverse Functions[Site to do: Rewrite this page]In general, if a pair of functions can be computed using the horizontal and vertical line rules with the set S in the plane then each function inverts or undoes the other.
A function y =f(x) which is injective on its domain has an inverse. The latter may be obtained from the graph of y = f(x) using the horizontal line method. The graph of the inverse is the transpose or reflection across the line y = x of the graph of y = f(x)>
For S = graph(f)
Exercises: Show the following:
Remark 1. The natural logarithm ln(x) may be obtained as the inverse function for the (natural) exponential function exp(x) = ex, and vice-versa. The graph of each is the reflection of that of the other across y = x. Remark 2. Inverses of trig functions (sine, cosine, tangent) and so on are obtain by domain restrictions that yield sets with the horizontal line properties. Which domain restriction to take may be a matter of convenience or convention. Read the manual for your calculator to determine how those inverses are defined. Remark 3. With coordinates in the plane, we can describe or represent computation rules (functions) in standard and non-standard ways. The standard way puts the dependent variable first and independent variable second. Doing so gives the graph of the function f. The vertical line rule gives a means for finding the dependent variable y = f(x) from an the independent variable x. The non-standard way puts the dependent variable second and the independent variable first. Doing so provides a non-standard graph of the function x = h(y) - the standard graph reflected across the line y = x. That being said, the horizontal line rule gives a method for calculating x = h(y) from the independent variable y. More on Inverse Functions and Their Calculation/DefinitionThe set or curve in the plane viewpoint (Route 2) has advantages in discussing the backward use of formulas y = f(x) where instead of calculating or obtaining y from x as in the forward use, we try to obtain x from y. Remember that when you meet the discussion of inverse functions. A curve in the plane may be regarded as a set of points or ordered pairs. The graph of a function f even when the function f or f(x) is introduced by other means ; may be used for calculation of y = f(x), that is the forward use of the function, and for the backward question of how x depends on y when y = f(x), y is given and x is to be computed. This backward question provides a context for the following.
The foregoing lessons provide a basis for defining inverse trigonometry using parts of the graphs of trig functions - the restriction of the latter to intervals to obtain functions that are one-to-one (invective) The twist, reflection across the line y = x in the Cartesian plane, connects the graph of a function and the graph of its inverse. In calculus, the area under the curve definition of
the natural logarithm leads to a one-to-one function. Its inverse is
the exponential function. Algebraic Calculation of Inverse Functions. Suppose y = f(x) where f(x) is a function given by a formula of some type. The inverse functionf--1(x) = g(x) if it exist, should have the property that g(f(x)) = x for each x in domain of f and also f(g(w)) = w for each w in the range of the original function f. Now f(y) = x may imply y = h(x) for some unique function h(x) or it may give more than one formula or solution h(x) for y. In the latter case, the function f is not one to one. In the former case, f is one to one, f--1 exists, and f--1(x) = h(x). Proof that f--1(x) = h(x). : If x = f(h(x)) then by substitution f--1(x) = f--1( f( h(x)) = f--1(f(y)) = y = h(x) Remark: If f(y) = x implies an equation linear in y (with the y coefficient nonzero) then y will be uniquely determined. If f(y) = x implies an equation quadratic in y (or more generally with a polynomial dependence on y) then their could 2 or more formulas h(x) for y, one formula per real root of a quadratic or more general polynomial in y. |
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Home < More Algebra < 4 Functions << 23 Inverse Functions
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