05Cinversefuncs

toc = Inverse Functions =

Recall that when we describe a function as ** many-to-one **, this refers to many x values having the same y-value.

A **one-to-many** relation is not a function (because one x value has many y-values, so it fails the vertical line test)

A **one-to-one** function will pass the vertical line test and the equivalent __horizontal line test__.

A ** one-to-one ** function that is __continuous__, can only be either:
 * a strictly increasing function (increasing over its entire domain) ** (OR) **
 * a strictly decreasing function (decreasing over its entire domain)

A function will have an inverse that is also a function if and only if it is **one-to-one**.

Notation
If a function, f(x) is a **one-to-one** function, then its **inverse function** is denoted by f –1 (x).

You are already familiar with this notation from sin –1 (x) (which is the inverse of sin(x))


 * WARNING **

Be careful not to confuse the notation for an inverse function f –1 (x) with the notation for the reciprocal of a function: math \left( f(x) \right)^{-1} = \dfrac{1}{f(x)} math

Maximal Domain
The ** maximal domain ** (or **implied domain**) of a function is the largest domain for which its rule is defined. If a function is given without a domain, it is assumed that the maximal domain is being used.

You may be asked to find the largest domain for a function for which its inverse function is defined (for which f –1 (x) exists).

This usually implies that for the maximal domain, the inverse is not a function. You are being asked to state the largest restricted domain so that that section of the function is one-to-one (ie the inverse function, f –1 (x) exists).

** Example 1 **
State the largest negative domain of f(x) = (x + 3) 2, so that f –1 (x) exists. Hence fully define the restricted function, f(x) and its inverse function, f –1 (x).

Notice that the maximal (or implied) domain is x Î R. And that for the maximal domain, f(x) is __not__ one-to-one.

By observation of the graph, f(x) is one-to-one for two equally large domains: x Î (– ¥, –3] x Î [–3, + ¥ )

Hence, the largest __negative__ domain for which the inverse function exists is: x Î (– ¥, –3] and for this domain, the range is: y Î [0, + ¥ )

The inverse of f(x) can be found by swapping x and y in the rule: math \\ . \qquad y=(x+3)^2 \\ .\\ . \qquad \text{Reverse x and y} \qquad. \\ . \\ . \qquad x = (y+3)^2 \\.\\ . \qquad (y+3)^2 = x \\.\\ . \qquad y+3=\pm \sqrt{x} \\. \\ . \qquad y=\pm \sqrt{x} -3 math

but we want the range of f –1 (x) to be the domain of f(x): (– ¥, –3] so choose the negative half, hence: math f^{-1}(x)=-\sqrt{x}-3 math

Hence, fully defined, the function and its inverse function are: math \\ f:(-\infty,-3] \rightarrow R, f(x)=(x+3)^2 \\ \\ f^{-1}: [0, +\infty) \rightarrow R, f^{-1}(x)=-\sqrt{x}-3 math

Key Feature of Inverse Functions
A feature of inverse functions is that the composite of a function and its inverse will always give the variable (x) as the result. math \\ f \left( f^{-1}(x) \right) = x \\ \\ f^{-1} \left( f(x) \right) = x math

** Example 2 **
a) sin(sin –1 (x)) = x

b) sin –1 (sin(x)) = x

c) log e (e x ) = x

d) e log e (x) = x

e) ** {using f(x) from example 1, above} ** math \\ f(x)=(x+3)^2 \\ \\ f^{-1}(x)=\sqrt{x}-3 math

math f \left( f^{-1}(x) \right) = \big( \left( \sqrt{x}-3 \right) + 3 \big)^2 math math \\ . \qquad \qquad \; \; = \big( \sqrt{x} \big)^2 \\ \\ . \qquad \qquad \; \; = x math

f) math \\ f^{-1}(x) = \sqrt{x}-3 \\ \\ f^{-1} \big( f(x) \big) = \sqrt{(x+3)^2} - 3 \\ \\ . \qquad \qquad \; \; = (x + 3) - 3 \\ \\ . \qquad \qquad \; \; = x math Go to top of page flat

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