What is a Function - Pre-Calculus

1. What is a Function
Chapter 1
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2. A function 𝑓 is a rule that assigns to each element x in a
set 𝐷 exactly one element, called 𝑓 𝑥 , in a set of 𝐸.
• We usually consider functions for which the sets 𝐷 and 𝐸 are sets of
real numbers.
• The set 𝐷 is called the domain of the function.
• The range of 𝑓(𝑥) is the set of all possible values of 𝑓(𝑥) as 𝑥 varies
throughout the domain.
• A symbol that represents an arbitrary number in the domain of a
function 𝑓 is called an independent variable.
• A symbol that represents a number in the range of 𝑓 is called a
dependent variable.
2

3. THE VERTICAL LINE TEST A curve in the -plane is the graph of a function
of if and only if no vertical line intersects the curve more than once.
3

4. Example 1.1
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Is it a Function?

5. Even and Odd Functions
• If a function 𝑓 satisfies 𝑓 𝑥 = 𝑓(−𝑥) for every number in its domain, then 𝑓
is called an even function.
• For example, 𝑓 𝑥 = 𝑥2
is even because 𝑓 −𝑥 = (−𝑥)2
= 𝑥2
= 𝑓(𝑥)
• If 𝑓 satisfies 𝑓 −𝑥 = −𝑓(𝑥) for every number in its domain, then 𝑓 is called
an odd function.
• For example, 𝑓 𝑥 = 𝑥3
is odd because 𝑓 −𝑥 = (−𝑥)3
= −𝑥3
= −𝑓(𝑥)
5

6. Example 1.1
a) 𝑓 𝑥 = 𝑥5
+ 𝑥
6
Determine whether each of the
following functions is even, odd, or
neither even nor odd.
b) 𝑔 𝑥 = 1 − 𝑥4
c) ℎ 𝑥 = 2𝑥 − 𝑥2

7. Example 1.2.a
𝑓 −𝑥 = −𝑥5 + −𝑥 = (−1)5𝑥5 + −𝑥
= −𝑥5 − 𝑥 = − 𝑥5 + 𝑥
𝑓 −𝑥 = −𝑓 𝑥
Therefore, 𝑓 is an odd function.
7

8. Example 1.2.b
𝑔 −𝑥 = 1 − (−𝑥)4
= 1 − 𝑥4
𝑔(−𝑥) = 𝑔(𝑥)
Therefore, 𝑔 is an even function.
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9. Example 1.2.c
ℎ −𝑥 = 2 −𝑥 − (−𝑥)2
= −2𝑥 − 𝑥2
𝑓 −𝑥 ≠ 𝑓 𝑥 𝑎𝑛𝑑 𝑓 −𝑥 ≠ −𝑓 𝑥
Therefore, ℎ is neither even or odd.
9
Notice that the graph of h is symmetric
neither about the y-axis nor about the
origin.

10. 10

11. Vertical Shift:
y = 𝑓 𝑥 + 𝑘
• Shift the graph of 𝑓 up
k units k > 0
• Shift the graph of 𝑓
down 𝑘 units k < 0
Horizontal Shift:
y = 𝑓 𝑥 + ℎ
Shift the graph of 𝑓
left h units k > 0
Shift the graph of 𝑓
right 𝑘 units k < 0
11

13. Stretching
𝑦 = 𝑐𝑓 𝑥
Stretch the graph
vertically by c
𝑦 = 𝑓
𝑥
𝑐
Stretch the graph
horizontally by c
Compressing
𝑦 =
1
𝑐
𝑓 𝑥
Compress the graph
vertically by c
𝑦 = 𝑓 𝑐𝑥
Compress the graph
horizontally by c
Reflection
𝑦 = −𝑓 𝑥
Reflects the graph across
the x-axis
𝑦 = 𝑓 −𝑥
Reflects the graph across
the y-axis
13

14. Assignment
1.1
1.1.1. 𝑠𝑙𝑜𝑝𝑒 = −6, 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 1,3
1.1.2.𝑠𝑙𝑜𝑝𝑒 = 3, 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ −3,2
1.1.3.𝑠𝑙𝑜𝑝𝑒 =
1
3
, 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 0,4
1.1.4.𝑠𝑙𝑜𝑝𝑒 =
2
5
, 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 8
1.1.5. 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 2,1 𝑎𝑛𝑑 −2, −1
1.1.6. 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ −3,7 𝑎𝑛𝑑 1,2
1.1.7. 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 5, 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −3
1.1.8. 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −6, 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 9
14
write the equation of the line satisfying the
given conditions in slope-intercept form of the
following expressions

15. Assignment
1.2
1.2.1. 𝑓 𝑥 = 2𝑥5 − 3𝑥2 + 2
1.2.2. 𝑓(𝑥) = 𝑥3
− 𝑥7
1.2.3. 𝑓(𝑥) = 𝑒−𝑥2
1.2.4. 𝑓(𝑥) = 1 + 𝑠𝑖𝑛𝑥
15
Determine whether 𝑓 is even, odd, or
neither even nor odd.