2. The Function Rule
โข A function can be thought of as a rule which
operates on an input and produces and output.
3. The Function Notation
๐ ๐ฅ = 3๐ฅ
โข Read as โ๐ as a function of ๐โ or โ๐ of ๐โ or โthe rule of ๐ isโฆ.โ
โข meaning that the value of the output from the function depends upon the value of the
input x.
โข The value of the output is often called the โvalue of the functionโ.
Name of the
function
Input Output (value of the function)
4. The Function Notation
Given the following functions, find the value of the output with each given
input or argument:
1. Given: ๐ ๐ฅ = 3๐ฅ + 1
i. ๐ 2
ii. ๐ โ1
iii. ๐(6)
2. Given: ๐ฆ ๐ฅ = 3๐ฅ + 2
i. ๐ฆ ๐ก
ii. ๐ฆ 2๐ก
iii. ๐ฆ(๐ง + 2)
5. The Function Notation
Given the following functions, find the value of
the output with each given input or argument:
3. ๐ ๐ฅ = 3๐ฅ2 โ 7
i. ๐(3๐ก)
ii. ๐ ๐ก + 5
6. Operations on Functions
โขSimilar to any real numbers, functions can also be added,
subtracted, multiplied and divided.
EQUATION OPERATION
๐ + ๐ ๐ = ๐ ๐ + ๐(๐) Addition
๐ โ ๐ ๐ฅ = ๐ ๐ฅ โ ๐(๐ฅ) Subtraction
๐ โ ๐ ๐ฅ = ๐(๐ฅ) โ ๐(๐ฅ) Multiplication
๐
๐
๐ฅ = ๐(๐ฅ)/๐ ๐ฅ Division
๐ โ ๐ ๐ฅ = ๐(๐ ๐ฅ ) Composition
7. Operations on Functions
Examples:
1. Given ๐ ๐ฅ = 3๐ฅ + 2 and ๐ ๐ฅ = 4 โ 5๐ฅ,
Find:
i. (๐ + ๐)(๐ฅ)
ii. ๐ โ ๐ ๐ฅ
iii. ๐ โ ๐ ๐ฅ
iv.
๐
๐
๐ฅ
8. Operations on Functions
Examples:
2. Given: ๐ ๐ฅ = 2๐ฅ, ๐ ๐ฅ = ๐ฅ + 4, and โ ๐ฅ = 5 โ ๐ฅ3
.
Find:
i. ๐ + ๐ 2
ii. โ โ ๐ 2
iii. ๐ โ โ 2
iv.
โ
๐
2
v. (๐ โ โ)(2)
9. Drills
For numbers 5 and 6, evaluate the following functions.
5. Given the function ๐ ๐ฅ = 5๐ฅ + 8, find the value of the
output given the following input or argument
a. ๐ ๐ก
b. ๐ โ8
c. ๐ 5
d. ๐ 3๐ฅ
e. ๐(7 โ ๐ฅ)
10. Functions in Real-Life Situations
Example 1:
f(t) = 10,000(1.05)t represent the amount of money if P10,000 is invested at 5% compounded
annually.
Example 2:
Suppose that a jeepney ride costs P8 for the first 4 kilometers, then an additional P1.50 for each
succeeding kilometer.Then the cost of a jeepney fare for a jeepney on an 8-kilometer route can be
modeled by:
๐น( ๐) =
8.00 ๐๐ 0 < ๐ โค 4
9.50 ๐๐ 4 < ๐ โค 5
11.00 ๐๐ 5 < ๐ โค 6
12.50 ๐๐ 6 < ๐ โค 7
14.00 ๐๐ 7 < ๐ โค 8
where d is the distance travelled and F(d) is the fare.
11. Piecewise Function
โข Is a function defined by multiple sub-functions with each sub-
function applying to a certain interval of the functionโs domain
(or a sub-domain)
๐น( ๐) =
8.00 ๐๐ 0 < ๐ โค 4
9.50 ๐๐ 4 < ๐ โค 5
11.00 ๐๐ 5 < ๐ โค 6
12.50 ๐๐ 6 < ๐ โค 7
14.00 ๐๐ 7 < ๐ โค 8
12. Piecewise Function
Example:
โขThe fee for hiring a guide to hike Mt. Apo is P700. Suppose that
the guide can take care of at least 8 hikers and a maximum of 16.
For every succeeding 2 hikers, there is an additional fee of P125.
Represent the cost of hiring guides as a piecewise function of the
number of hikers.