2. Learning
Objectives:
At the end of the lesson you will
be able to:
- Determine the inverse of a one- to-
one functions.
- Represent an inverse function
through it’s: table of values and
graphs.
- Find the domain and range of an
inverse function.
3. What is a Inverse
Function?
- A set of ordered pairs formed by
reversing the coordinates of each
ordered pair of the function.
- Remember that function ‘’ f ‘’ has
an inverse if and only if ‘’ f ‘’ is one-
to- one function.
NOTE: Not every function has an
inverse.
4. Domain and Range of Inverse
Function:
- The domain of the inverse function
is the range of the function, and the
range of the inverse function is the
domain of the function.
5. Concept
check:
- A function y = f(x) is one- to- one if a
horizontal line drawn through the graph
of the function intersects the graph at
exactly one point.
- If the horizontal line intersects the
graph in more than one point, then the
function is not one- to- one. This test is
called geometric test for a one- to- one
6. Concept
check:
A function y = f(x) is said to be one-
to- one if:
− 𝑎 ≠ 𝑏 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑓 𝑎 ≠ 𝑓 𝑏 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟
𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓.
− 𝑇ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑓 𝑎𝑛𝑑 𝑓−1𝑎𝑟𝑒
Symmetric with respect to the line y = x.
7. One- to- one
Function:
Recall that a
function is a set
of ordered pairs
in which no two
ordered pairs
have the same x
and have
18. Steps in solving Inverse of a One- to-
one Function:
STEP 1: Replace f (x) by y.
STEP 2: Interchange x and y.
STEP 3: Solve for y in terms of x.
STEP 4: Replace y with 𝒇−𝟏
𝒙 .
19. Example 1: 𝐟 𝒙 = 𝟐𝒙 + 𝟒
STEP 1: Replace f
(x) by y.
𝒚 = 𝟐𝒙 + 𝟒
STEP 2: Interchange
x and y.
𝒙 = 𝟐𝒚 + 𝟒
STEP 3: Solve for y
in terms of x.
(−𝟐𝒚 = −𝒙 + 𝟒)(−𝟏)
𝟐𝒚 = 𝒙 − 𝟒)
𝟐
𝒚 =
𝟏
𝟐
𝒙 − 𝟐
STEP 4: Replace y with
𝒇−𝟏 𝒙 .
𝒇−𝟏
𝒙 =
𝟏
𝟐
𝒙 − 𝟐