1) The document discusses properties of equality and inverse functions. It provides examples of finding the inverse of functions like f(x) = 3x + 6. 2) Students are asked to find the inverse of various functions through a series of examples and activities. This includes making tables, graphing functions and their inverses, and determining domains and ranges. 3) One example finds the inverse of the function y = 150 + 50x, which describes a math tutor's hourly earnings. The inverse represents the number of students that can be assisted in an hour given a particular earnings amount.

1. Entry Card
For each operation in the first
column, record the inverse operation
in the second column.
Operation Inverse
1. Add 8
2. Subtract 10
3. Multiply by 5
4. Divide by 6
Subtract 8
Add 10
Divide by 5
Multiply by 6

2. Properties of Equality
Let x, y, and z are any real number.
1. Addition Property of Equality (APE)
If x = y, then x + z = y + z.
Equal values may be added on both sides
of the equation.
2. Subtraction Property of Equality (SPE)
If x = y, then x – z = y – z.
Equal values may be subtracted on both
sides of the equation.

3. 3. Multiplication Property of Equality (MPE)
If x = y, then xz = yz.
Both sides of the equation may be
multiplied by the same value.
4. Division Property of Equality (DPE)
If x = y, then z ≠ 0, then
𝑥
𝑧
=
𝑦
𝑧
.
Both sides of the equation may be
divided by the same non-zero
real number.
5. Substitution Law
If x + y = z and x = y, then y + y = z or
x + x = z.
Equals may be substituted for equals.

7. A = {(1, 3), (2, 4), (3, 5), (4, 6)}
B = {(3, 1), (4, 2), (5, 3), (6, 4)}
L
O
V
E
H
A
T
E
x y
L
O
V
E
H
A
T
E
x y

8. A = {(1, 2), (2, 3), (3, 4), (4, 2)}
B = {(2, 1), (3, 2), (4, 3), (2, 4)}
J
O
Y
S
O
R
R
O
W
x y
J
O
Y
S
O
R
R
O
W
x y

9. Definition:
If f is a one-to-one function,
then the inverse of f denoted
by f-1 is the function formed
by reversing all the ordered
pairs in f. Thus,
f-1 = (𝒚, 𝒙) 𝒙, 𝒚 𝒊𝒔 𝒊𝒏 𝒇

10. Properties of an
Inverse Function
If the f-1 inverse function exists,
1. f-1 is a one-to-one function,
f is also one-to-one.
2. Domain of f-1 = Range of f.
3. Range of f-1 = Domain of f.

11. Example:
Find the inverse of the function
f(x) = 3x + 6.
Solution:
f(x) = 3x + 6
y = 3x + 6
x = 3y + 6
3y = x – 6
y =
1
3
𝑥 − 2
f-1 =
1
3
𝑥 − 2
f(x) = y
Interchange x and y.
Subtract both sides by 6.
Multiply both sides by 3.
Replace the new y with f-1.

12. Boardwork:
Find the inverse function of the
following:
1. f(x) = x - 2
2. f(x) = 𝑥, x ≥ 0
3. f(x) = 5x + 3
4. f(x) = x2 + 3, x ≥ 0
5. f(x) = x3

13. Directions:
1. Make a table of values and graph the
given function.
2. Find its inverse function, make a
table of values and graph in the
same coordinate plane.
Group 1: f(x) = x + 4
Group 2: f(x) = ∣x - 2∣, x ≥ 0
Group 3: f(x) = x3 + 1
Group 4: f(x) = (x + 4)2, x ≥ 0
Group 5: f(x) = 2𝑥 + 1, x ≥ 0
Activity 1

14. Activity 2
Directions:
For each function f below,
a. Determine f-1(x).
b. State the domain and range of both
f and f-1.
c. Verify that f-1(f(x)) = x for each x in
the domain of f and f(f-1(x)) = x for
each x in the domain of f-1.
d. Draw the graphs of f and f-1 in the
same coordinate plane and state
their characteristics.

15. 1. f(x) = 5x + 3
2. f(x) = -
3𝑥
𝑥+7
3. f(x) = x2 + 3, x ≥ 0
4. f(x) = x3 – 2
5. f(x) = 1 - 2 𝑥, x ≥ 0

16. Activity 3
The function y = 150 + 50x describes
the hourly wage (y) of a math tutor
earnings a flat fee of Php 150 plus
Php 50 for each student the tutor
assists during an hour. Find the
inverse of this function. Write a
sentence or two telling what this
inverse describes, discussing what
each variable in the inverse function
represents.