Algebra 2 Unit 9:
Functional Relationships
Topic: Functions & Their Inverses
Vocabulary

Inverse Relation
– A relation that “undoes” a function.
– The domain of a function is the range of its
inverse; the range of a function is the domain of
its inverse.
– The graphs of a function & its inverse are
symmetric about the line y = x.

One-to-one Function
– A function in which each range value is paired
with one and only one domain value.
– If a function, f is one-to-one, then it’s inverse is
also a one-to-one function and is notated f -1.
Determining whether a function is oneto-one

If a function passes the horizontal line test,
it is a one-to-one function.
– Any horizontal line must pass through the graph of
a function once and only once.
Function is one-toone. Inverse will also
be a function.
Function is not oneto one. Inverse will
not be a function.
Finding the inverse of a function
f ( x)   x  2
2
1
2
Replace f (x) with y, then switch x & y in the
equation.
y   x  2
1
2
2
1
2
2
x   y  2
Solve the resulting equation for y.
 x  12 y  2
Take the square root of both sides
(remember there are two solutions).
 x  2  12 y
Subtract 2 from both sides.
2(  x  2 )  y
Multiply both sides by 2.
y  2 x  4
The resulting relation is the inverse of f (x).
Inverse Functions

Determine if the given function is one-to-one. If so,
find its inverse & state its domain & range.
f ( x)   x  2
1
2
2
The graph of the function does not pass the
horizontal line test. It is not one-to-one,
therefore its inverse is not a function (and
we’re done with this problem!)
Inverse Functions

Determine if the given function is one-to-one. If so,
find its inverse & state its domain & range.
3
f ( x) 
x2
The graph of the function does pass the
horizontal line test. It is one-to-one, therefore
its inverse is a function, and we must find it.
Inverse Functions

Determine if the given function is one-to-one. If so,
find its inverse & state its domain & range.
3
Replace f (x) with y and switch x & y.
f ( x) 
x2
y 
3
x2
x
3
y2
x  y2
3
Solve for y to find the inverse.
Since we know this is a function, we must notate it
properly (change y to f -1).
y  x 2
3
f
1
( x)  x  2
3
The domain & range of f (x) is all real #s, thus
the domain & range of f -1(x) is all real #s.
Homework
Quest: Functions & Their Inverses
Due 5/7 (A-day) or 5/8 (B-day)