6.1. Inverse Functions http://www.ck12.org
How to find the inverse of a one-to-one function:
To find the inverse of a one-to-one function, simply solve forxin terms ofyand then interchangexandy.The
resulting formula is the inversey=f−^1 (x).
Example 3:
Find the inverse off(x) =√ 4 x+1.
Solution:
From the discussion above, we can find the inverse by first solving forxiny=√ 4 x+1.
y=√ 4 x+ 1 ,
y^2 = 4 x+ 1 ,
x=y(^2) − 1
4.
Interchangingx←→y,
y=x
(^2) − 1
4.
Replacingy=f−^1 (x),
f−^1 (x) =x
(^2) − 1
4
which is the inverse of the original functionf(x) =√ 4 x+1.
Graphs of Inverse Functions
What is the relationship between the graphs offandf−^1? If the point(a,b)is on the graph off(x),then from the
definition of the inverse, the point(b,a)is on the graph off−^1 (x).In other words, when we reverse the coordinates
of a point on the graph of f(x)we automatically get a point on the graph of f−^1 (x).We conclude that f(x)and
f−^1 (x)arereflectionsof one another about the liney=x.That is, each is a mirror image of the other about the line
y=x.The figure below shows an example ofy=x^2 and, when the domain is restricted, its inversey=√xand how
they are reflected abouty=x.