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Section 4.1: Inverses

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If the functions f and g satisfy two conditions:

Section 4.1: Inverses

g(f(x)) = x for every x in the domain of f

f(g(x)) = x for every x in the domain of g

then f and g are inverse functions. f is an inverse of g and g is an inverse of f

Determine whether f and g are inverses:

Section 4.1: Inverses

f and g are inverses

Method for determining the inverse of a function:

Section 4.1: Inverses

Solve for x:

A function f has an inverse if and only if its graph is cut at most once by any horizontal line:

Section 4.1: Inverses

A function which passes the vertical and horizontal line tests has an inverse and is one-to-one.

Graphically:

Section 4.1: Inverses

If f has an inverse, the graphs of the functions are reflections of one another about the line y = x.

If the graph of f is always increasing or decreasing, then the function f has an inverse.

Section 4.1: Inverses

How can we show a function is always increasing or decreasing?

The domain of the original is the range of the inverse. The range of the original is the domain of the inverse.

Section 4.1: Inverses