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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:.

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section 4 1 inverses

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

section 4 1 inverses3

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.

section 4 1 inverses4

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.

section 4 1 inverses5

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?

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