6.4A Use Inverse Functions
Algebra II
Review from chapter 2
• Relation – a mapping of input values (x-values)
onto output values (y-values).
• Here are 3 ways to show the same relation.
y=
x2
Equation
Table of
values
Graph
x
y
-2
4
-1
1
0
0
1
1
• Inverse relation – just think: switch the x & y-values.
x = y2
x
y
y x
4
-2
1
-1
0
0
1
1
** the inverse
of an
equation:
switch the x
& y and
solve for y.
** the
inverse of a
table:
switch the
x & y.
** the inverse of a
graph: the reflection
of the original graph
in the line y = x.
Ex. 1: Find an inverse of y = -3x+6.
• Steps: -switch x & y
-solve for y
y = -3x+6
x = -3y+6
x-6 = -3y
x6
y
3
1
y  x2
3
Inverse Functions
• Given 2 functions, f(x) & g(x), if f(g(x))=x
AND g(f(x))=x, then f(x) & g(x) are
inverses of each other.
Symbols: f -1(x) means “f inverse of x”
Ex. 2: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are
inverses.
• Meaning find f(g(x)) and g(f(x)). If they both
equal x, then they are inverses.
f(g(x))= -3(-1/3x+2)+6
g(f(x))= -1/3(-3x+6)+2
= x-6+6
= x-2+2
=x
=x
** Because f(g(x))=x and g(f(x))=x, they are inverses.
6.4B
To find the inverse of a function:
Ex 1: (a)Find the inverse of f(x)=x5.
(b) Is f -1(x) a function?
1. y = x5
2. x = y5
3. 5 x  5 y 5
5
xy
y x
5
Yes , f -1(x) is
a function.
(hint: look at the graph!
Does it pass the vertical line
test?)
Horizontal Line Test
• Used to determine whether a function’s
inverse will be a function by seeing if the
original function passes the horizontal
line test.
• If the original function passes the
horizontal line test, then its inverse is a
function.
• If the original function does not pass the
horizontal line test, then its inverse is not
a function.
Ex. 2: Graph the function f(x)=x2 and
determine whether its inverse is a
function.
Graph does not pass the
horizontal line test,
therefore the inverse is not
a function.
Ex 3: f(x)=2x2-4 Determine whether f -1(x) is
a function, then find the inverse equation.
y = 2x2-4
x = 2y2-4
x+4 = 2y2
x4
 y2
2
x4
y
2
f -1(x) is not a function.
OR, if you fix the
tent in the
basement…
1
y  x2
2
Ex. 4: g(x)=2x3
y=2x3
x=2y3
x
 y3
2
3
x
 y
2
y
Inverse is a function!
3
OR, if you fix the
tent in the
basement…
x
2
y
3
4x
2
Assignment