Verifying Inverses
Verifying –checking that something is
true or correct
Inverses – two equations where the x’s
and y’s of one equation are switched
in the second equation
How do we verify
inverses?
 There are two ways to determine if two
functions are inverses of each other:
1. Graphically – Are all the points of one graph
reflections over the line y=x from the other graph? (Corresponding
points have to switch the x and y coordinates.)
2. Algebraically – Do both compositions simplify to “x”?
Verifying Inverses
GRAPHICALLY
Is EVERY point on one graph a reflection
over the line y = x of a point on the other
graph?
Do they cross the line y = x at the same
place?
Yes, they are INVERSES since all points
are reflections over y = x !!! And they
intersect on the “mirror” line.
Verifying Inverses
GRAPHICALLY
Are the lines
reflections over y = x?
Do they intersect on
the line y = x?
No. They do not intersect on the
“mirror” line. Nor do the points
reflect over y = x! Ex: (4,10) does
not go to (10,4)
Verifying Inverses
ALGEBRAICALLY
 If functions f and g are inverses of
each other,
then f(g(x)) = x and g(f(x)) = x.
 What this means is:
if we substitute one function’s equation in for the x
of the other equation and then simplify the
expression, we’ll be left with just ‘x’.
*** f(g(x)) and g(f(x)) are called compositions of
functions.
Verifying Inverses
ALGEBRAICALLY
Find both compositions… Plug each function into the other and simplify.
f(g(x)) = 2(1/2x – 2) + 4
g(f(x)) = ½ (2x + 4) – 2
= x – 4 + 4 … distribute
= x + 2 – 2… distribute
=x
=x
… this one is good.
… both are good
Since both compositions simplify to “x” they are
inverses of each other.
Extra Example
Find both compositions… Plug each function into the other and simplify.
f(g(x)) = 4(1/4x + 4) – 16
g(f(x)) = ¼ (4x – 16) + 4
= x + 16 – 16 … distribute
= x – 4 + 4… distribute
=x
=x
… this one is good.
… both are good
Since both compositions simplify to “x” they are
inverses of each other.
Verifying Inverses
Algebraically and Graphically
f(g(x)) = 4(1/4x – 3) + 3
= x – 12 + 3 … distribute
= x – 9 … this one is
NOT good so they cannot
be inverses
g(f(x)) = ¼ (4x + 3) – 3
= x + 3/4 – 3
= x – 2.25 …
neither one is
good
If either one of the
compositions does NOT
simplify to “x” then they are
NOT inverses of each other.
GRAPHICALLY…
They do NOT intersect on the
line y = x so they are NOT
inverses. (Points are NOT
reflected across y = x either.)
Verifying Inverses
Graphically and Algebraically
Try #5 on your own, and then you can
check your work by moving on to the
next two slides.
 The first slide will show the problem
done algebraically.
 The second slide will show the
problem done graphically.
#5 –Algebraically
#5 - Graphically