(𝑓 ∘ 𝑔)(𝑥) = (𝑥 2 + 𝑥) + 1
(𝑓 ∘ 𝑔)(𝑥) = 𝑥 2 + 𝑥 + 1
GENERAL MATHEMATICS
1st Quarter Reviewer
First Semester
FUNCTIONS
Function is a rule of correspondence between two
nonempty sets, such that, each element of the first set
corresponds to one and only one element of the second
set.
Vertical Line Test is used to determine if the graph is a
function or not.
Examples:
1) {(5,15), (6,16), (7,17), (8,18)}
2) {(−4, −3), (−2, −3), (0,1), (2,3), (4,3)}
3) {(−6,12), (−3,6), (0,5), (3,6), (6,12)}
Evaluation of Functions
✓ Evaluating a function means finding the value
of 𝑓(𝑥) or 𝑦 that corresponds to a given value of
𝑥.
✓ Steps in Evaluating a Function
Find 𝑓(1) if 𝑓(𝑥) = 2𝑥 2 + 𝑥 − 3.
1st: Substitute the given value of 𝑥
𝑓(1) = 2(1)2 + 1 − 3
2nd: Simplify
𝑓(1) = 2 + 1 − 3
𝑓(1) = 0
Operations on Functions
Perform the following fundamental operations given
𝑓(𝑥) = 𝑥 + 1 and 𝑔(𝑥) = 𝑥 2 + 𝑥
✓ (𝒇 + 𝒈)(𝒙) = 𝒇(𝒙) + 𝒈(𝒙)
(𝑓 + 𝑔)(𝑥) = (𝑥 + 1) + (𝑥 2 + 𝑥)
(𝑓 + 𝑔)(𝑥) = 𝑥 2 + 2𝑥 + 1
✓ (𝒈 ∘ 𝒇)(𝒙) = 𝒈[𝒇(𝒙)]
(𝑔 ∘ 𝑓)(𝑥) = (𝑥 + 1)2 + (𝑥 + 1)
(𝑔 ∘ 𝑓)(𝑥) = 𝑥 2 + 2𝑥 + 1 + 𝑥 + 1
(𝑔 ∘ 𝑓)(𝑥) = 𝑥 2 + 3𝑥 + 2
Solving Problems Involving Function
The cost of producing x gadget by a company is given (in
pesos) by the function 𝑓(𝑥) = 1,500𝑥 + 7,000. What is
the cost of producing 50 gadgets? 100 gadgets?
✓ Understand the Problem
Given: 𝑓(𝑥) = 1500𝑥 + 7000
50 gadgets
100 gadgets
Task: Find the cost of producing 50 gadgets and
100 gadgets
✓ Devise a Plan
Substitute the number of gadgets as the value of
𝑥 in the given function to find the cost
✓ Carry out the Plan
𝑓(50) = 1,500(50) + 7,000
𝑓(50) = 82,000
𝑓(100) = 1,500(100) + 7,000
𝑓(100) = 157,000
✓ Review the Solution
RATIONAL FUNCTIONS
Real-life Applications of Rational Functions
✓ Speed
✓ Amount of money earned and hours of work
✓ Population growth rate
Rational Function is a function of the form 𝑓(𝑥) =
𝑔(𝑥)
ℎ(𝑥)
✓ (𝒇 − 𝒈)(𝒙) = 𝒇(𝒙) − 𝒈(𝒙)
(𝑓 − 𝑔)(𝑥) = (𝑥 + 1) − (𝑥 2 + 𝑥)
(𝑓 − 𝑔)(𝑥) = 𝑥 + 1 − 𝑥 2 − 𝑥
(𝑓 − 𝑔)(𝑥) = −𝑥 2 + 1
where 𝑔(𝑥) and ℎ(𝑥) are polynomial functions and ℎ(𝑥)
is not equal to 0.
✓ (𝒇 ∙ 𝒈)(𝒙) = 𝒇(𝒙) ∙ 𝒈(𝒙)
(𝑓 ∙ 𝑔)(𝑥) = (𝑥 + 1)(𝑥 2 + 𝑥)
(𝑓 ∙ 𝑔)(𝑥) = 𝑥 3 + 𝑥 2 + 𝑥 2 + 𝑥
(𝑓 ∙ 𝑔)(𝑥) = 𝑥 3 + 2𝑥 2 + 𝑥
✓
𝒇
𝒈
𝑓
𝒇(𝒙)
𝒈(𝒙)
𝑥+1
𝑔
𝑓
𝑥 2 +𝑥
𝑥+1
✓ ( ) (𝒙) =
( ) (𝑥) =
( ) (𝑥) =
𝑔
𝑥(𝑥+1)
𝑓
1
( ) (𝑥) =
𝑔
𝑥
✓ (𝒇 ∘ 𝒈)(𝒙) = 𝒇[𝒈(𝒙)]
Rational
Inequality
Rational Equation
2𝑥
4
= − 2𝑥
15
5
𝑥
4
✓
=
−3
2−𝑥
𝑥+4
4𝑥+5
5𝑥
>
𝑥−4
𝑥−4
𝑥
2
✓
<3−
𝑥−2
𝑥−2
✓
Rational Function
𝑥 3 −1
𝑥+1
1
✓ 𝑔(𝑥) =
𝑥
✓ 𝑟(𝑥) =
Solving Rational Equation
2𝑥
3
Solve the rational equation
=
𝑥+4
2𝑥+8
1st: Eliminate denominators by multiplying each
term of the equation by the least common
denominator
2𝑥
3
(2𝑥 + 8) (
=
)
𝑥 + 4 2𝑥 + 8
2𝑥(2) = 3
4𝑥 = 3
3
𝑥=
4
2nd: Substitute the obtained value of 𝑥 in the
equation to check if it satisfies the equality
3
2(4)
3
+4
4
3
2
19
4
=
3
3
2(4)+8
3
= 38
4
3 4
4
( ) = 3( )
2 19
38
12
12
=
38
38
Solving Rational Inequality
4𝑥+4
Solve the rational inequality
>3
𝑥
st
1 : Rewrite the inequality as a single rational
expression on one side of the inequality symbol
and 0 on the other side.
4𝑥+4
−3>0
𝑥
4𝑥+4−3(𝑥)
>0
𝑥
𝑥+4
>0
𝑥
nd
2 : Determine over what intervals the rational
expression takes on positive and negative values.
a. Locate the x values for which the rational
expression is zero or undefined (factoring
the numerator and denominator is a useful
strategy).
𝑥+4= 0
𝑥=0
𝑥 = −4
b. Mark the numbers found in (a) on a
number line. Use a shaded circle to indicate
that the value is included in the solution set,
and a hollow circle to indicate that the
value is excluded. These numbers partition
the number line into intervals.
(−∞, −4) (−4,0)
(0, ∞)
c. Select a test point within the interior of
each interval in (b). The sign of the rational
expression at this test point is also the sign
of the rational expression at each interior
point in the interval.
d. Summarize the intervals containing the
solutions.
{𝑥|𝑥 < −4, 𝑥 > 0} 𝑜𝑟 (−∞, −4) ∪ (0, ∞)
Intercepts of a Rational Function
2𝑥+2
Find the x-intercept and y-intercept of 𝑓(𝑥) = 2
4𝑥 −4
✓ x-intercept
o 1st: Equate the numerator to zero
2𝑥 + 2 = 0
o 2nd: Factor the expression
No need to factor
o 3rd: Equate each factor to 0
2𝑥 + 2 = 0
2𝑥 = −2
𝑥 = −1
∴ (−1,0)
✓ y-intercept
o 1st: Let 𝑥 be equal to 0
2(0) + 2
𝑓(0) =
4(0)2 − 4
nd
o 2 : Simplify
2
𝑓(0) =
−4
1
𝑓(0) = −
2
1
∴ (0, − )
2
Asymptotes of a Rational Function
2𝑥+2
Find the x-intercept and y-intercept of 𝑓(𝑥) = 2
4𝑥 −4
✓ Vertical Asymptote
o Equate the denominator to zero
4𝑥 2 − 4 = 0
o Factor the expression
(2𝑥 + 2)(2𝑥 − 2) = 0
o Equate each factor to zero
2𝑥 + 2 = 0
2𝑥 − 2 = 0
2𝑥 = −2
2𝑥 = 2
𝑥 = −1
𝑥=1
o Make sure that the obtained x-value is not
an x-intercept of the function. If it is an xintercept, it is not considered as vertical
asymptote.
𝑥 = −1 is an x-intercept of the function
o State the vertical asymptote
𝑥=1
✓ Horizontal Asymptote
(DN is degree of numerator; DD is degree of
denominator)
o 𝐷𝑁 < 𝐷𝐷; HA is 𝑦 = 0
𝑎
o 𝐷𝑁 = 𝐷𝐷; HA is 𝑦 = 𝑛 (𝑎𝑛 and 𝑏𝑚 are
𝑏𝑚
the leading coefficients of numerator and
denominator)
o 𝐷𝑁 > 𝐷𝐷 (1 ℎ𝑖𝑔ℎ𝑒𝑟); No HA but has an
oblique asymptote
o 𝐷𝑁 > 𝐷𝐷 (𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛 1 ℎ𝑖𝑔ℎ𝑒𝑟);
HA and No oblique asymptote
∴ 𝐻𝐴 𝑖𝑠 𝑦 = 0
No
Domain and Range of a Rational Function
𝑥+2
Find the domain and range of 𝑓(𝑥) = 2
4𝑥 −4
✓ Domain
o 1st: Make sure that the function is in its
simplest form.
The function is in its simplest form
already.
nd
o 2 : Find the vertical asymptote(s) of 𝑓(𝑥)
by setting the denominator equal to zero.
4𝑥 2 − 4 = 0
(2𝑥 + 2)(2𝑥 − 2) = 0
2𝑥 + 2 = 0
2𝑥 − 2 = 0
2𝑥 = −2
2𝑥 = 2
𝑥 = −1
𝑥=1
o 3rd: Exclude the vertical asymptote(s)
obtained in the previous step from the set
of real numbers.
{𝑥|𝑥 ≠ −1,1}
✓ Range
o 1st: Make sure that the function is in its
simplest form.
The function is in its simplest form
already.
nd
o 2 : Find the horizontal asymptote of 𝑓(𝑥)
by comparing the degrees of the numerator
and the denominator.
HA is 𝑦 = 0
o 3rd: Exclude the horizontal asymptote(s)
obtained in the previous step from the set
of real numbers.
{𝑦|𝑦 ≠ 0}
Solving Problems Involving Rational Function
A movie rental store charges ₱250 initial fee, then ₱50 for
each movie you rent. What is the average cost function
and the average cost per movie if you rent 5 movies?
✓ Understand the Problem
Given: ₱250 initial fee
₱50 additional for each rented movie
Task: Determine the rational function which
represents the average cost and the average cost
if 5 movies were rented
✓ Devise a Plan
o Let 𝑥 be the number of rented movies
o Let 𝐶(𝑥) be the average cost per movie
o The charge of the movie rental store can
be represented as 250+50𝑥
✓ Carry out the Plan
o The average cost per movie can be
250+50𝑥
represented as 𝐶(𝑥) =
.
𝑥
o
If 5 movies were rented, the solution
would be:
250 + 50(5) 500
𝐶(5) =
=
= 100
5
5
The average cost per movie if 5 movies
were rented is ₱100
✓ Review the Solution
INVERSE FUNCTIONS
One-to-one Function
A function is one-to-one if every second element is paired
to only one first element.
Horizontal Line Test is used to determine if the graph is
a one-to-one function or not.
Inverse Function
A function 𝑔 is the inverse function of 𝑓 if the ordered
pairs of 𝑔 are the ordered pairs of 𝑓 written in reversed
order.
Note: A function has an inverse function if and only if it is
one-to-one.
Finding the Inverse of a Function
✓ Inverse of a Set of Ordered Pairs
Find the inverse of the function described by the
set of ordered pairs
{(3, −2), (1,0), (5,7), (4, −4), (10,2)}
o To find the inverse of the given function,
interchange the coordinates of each
ordered pair. If it is one-to-one, then the
given function has an inverse.
∴ {(−2,3), (0,1), (7,5), (−4,4), (2,10)}
✓ Inverse of a Function
𝑥
Find the inverse of 𝑓(𝑥) = + 5
5
o 1st: Change 𝑓(𝑥) to 𝑦
𝑥
𝑦 = +5
5
o 2nd: Interchange the variable 𝑥 and 𝑦
𝑦
𝑥 = +5
5
o 3rd: Solve for y in terms of 𝑥
𝑦
5 (𝑥 = + 5)
5
5𝑥 = 𝑦 + 25
5𝑥 − 25 = 𝑦
o 4th: Change 𝑦 from Step 3 to 𝑓 −1 (𝑥)
𝑓 −1 (𝑥) = 5𝑥 − 25
Representations of Inverse of a Function: Table of
Values and Graph
1
Construct the graph of 𝑓(𝑥) = 𝑥 + 5 and its inverse.
5
✓ Construct the table of values of the given
function
Choose at least 5 values of x. Substitute each
value of x in the give function to obtain the y
values.
x
-4
-2
0
2
4
y
4.2
4.6
5
5.4
5.8
✓ Construct the table of values of the inverse of
the function
Interchange the x values and y values for the
inverse
x
4.2
4.6
5
5.4
5.8
Y
-4
-2
0
2
4
✓ Graph the function and its inverse
Plot the ordered pairs of the function and its
inverse in the cartesian plane
✓ If 𝑘 and 𝑐 have the same signs, then there is no xintercept.
✓ The y-intercept of 𝑓(𝑥) is (0, 𝑘 + 𝑐).
Asymptotes of an Exponential Function
✓ The function has no vertical asymptotes.
✓ The horizontal asymptote of 𝑓(𝑥) is 𝑦 = 𝑐.
Domain and Range of an Exponential Function
✓ The domain of an exponential function is always
the set of all real numbers, {𝑥|𝑥 ∈ ℝ} or
(−∞, ∞).
✓ If 𝑘 is positive, then the range is 𝑦 > 𝑐 is
because the points of the graph will appear above
the asymptote.
✓ If 𝑘 is negative, then the range is 𝑦 < 𝑐 is
because the points of the graph will appear below
the asymptote.
Solving Problem Involving Exponential Function
EXPONENTIAL FUNCTIONS
Exponential Function
An exponential function is a function in the form 𝑓(𝑥) =
𝑘𝑎 𝑥 + 𝑐 where 𝑎 > 0, 𝑎 ≠ 1, 𝑘 ≠ 0 and is any real
number.
Two Trends of Exponential Function
✓ Exponential Growth
o In this trend, 𝑎 > 1 and 𝑓(𝑥) increases.
✓ Exponential Decay
o In this trend, 0 < 𝑎 < 1 and 𝑓(𝑥)
deacreases.
LOGARITHMIC FUNCTIONS
Logarithm
A logarithm is an exponent which 𝑏 must have to produce
𝑦. A logarithm is in the form log 𝑏 𝑦 = 𝑥 if and only if
𝑏 𝑥 = 𝑦 for 𝑏 > 0 and 𝑏 ≠ 1.
Note: Logarithmic function and exponential function are
inverses.
Changing Exponential to Logarithmic and Vice Versa
Solving an Exponential Equation
Solve the exponential equation 162𝑥 = 64𝑥+2
✓ 1st: Make the bases the same
(42 )2𝑥 = (43 )𝑥+2
44𝑥 = 43𝑥+6
nd
✓ 2 : Equate the exponents with each other
4𝑥 = 3𝑥 + 6
✓ 3rd: Solve for 𝑥
4𝑥 − 3𝑥 = 6
𝑥=6
Intercepts of an Exponential Function
✓ If 𝑘 and 𝑐 have opposite signs, then there is an xintercept.
✓
✓
Solving a Logarithmic Equation
Solve the logarithmic equation log 3 9 = 4𝑛 + 6
✓ 1st: Apply law of logarithm
No law of logarithm to be applied
✓ 2nd: Change the logarithmic equation into its
exponential form
34𝑛+6 = 9
✓ 3rd: Apply the one-to-one property of exponential
equation
34𝑛+6 = 32
th
✓ 4 : Since the bases are the same, then their
exponents are equal
4𝑛 + 6 = 2
✓ 5th: Solve for 𝑥
4𝑛 = 2 − 6
4𝑛 = −4
𝑛 = −1
Finding the Intercepts of a Logarithmic Function
Find the intercepts of 𝑦 = log 3 (3𝑥 + 9)
✓ x-intercept
o 1st: Let 𝑦 be equal to 0
0 = log 3 (3𝑥 + 9)
o 2nd: Express the logarithmic function to its
exponential form
30 = 3𝑥 + 9
rd
o 3 : Solve for 𝑥
1 = 3𝑥 + 9
1 − 9 = 3𝑥
−8 = 3𝑥
8
− =𝑥
3
o 4th: State the x-intercept
8
(− , 0)
3
✓ y-intercept
o 1st: Let 𝑥 be equal to 0
𝑦 = log 3 (3(0) + 9)
𝑦 = log 3 9
o 2nd: Express the logarithmic function to its
exponential form
3𝑦 = 9
rd
o 3 : Solve for 𝑦
3𝑦 = 32
Since the bases are the same, then the
exponents are equal.
𝑦=2
o 4th: State the y-intercept
(0,2)
Domain and Range of a Logarithmic Function
✓ The domain is the set of all positive real numbers
(0, ∞) 𝑜𝑟 {𝑥|𝑥 ∈ ℝ+ }.
✓ The range is the set of all real numbers
(−∞, ∞)𝑜𝑟 {𝑥|𝑥 ∈ ℝ}.