MAE101
Mathematics for Engineering
Part 1: Calculus
Session 2: Functions and Graphs
Lecturer: TS. Trịnh Hoàng Minh
FPT University, Quy Nhon AI Campus
8/2023
Chapter 1: Functions and Graphs
• Review of Functions
• Basic Classes of Functions
• Inverse Functions
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Review of Functions
Function, domain, and range
• A function 𝑓 consists of a set of
inputs, a set of outputs, and a
rule for assigning each input to
exactly one output.
• The set of inputs: domain of 𝑓
• The set of outputs: range of 𝑓
Example
• 𝑓 𝑥 = 2𝑥 + 1
• 𝑓 𝑥 : 1, 𝑓 1 = 6 ; 2, 𝑓 2 = 4 ; 3, 𝑓 3 = 2 ; 4, 𝑓 4 = 2
3𝑥 + 1 𝑥 ≥ 2
• 𝑓 𝑥 =ቊ 2
𝑥
𝑥<2
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Function, domain, and range
• Example Evaluating the function 𝑓 𝑥 = 3𝑥 2 + 2𝑥 − 1 at
𝑓 −2 , 𝑓 2 , 𝑓 𝑎 + ℎ
• Example Finding domain and range of the following
functions
a) 𝑓 𝑥 = 𝑥 − 4 2 + 5
b) 𝑓 𝑥 = 3𝑥 + 2 − 1
c) 𝑓 𝑥
3
=
𝑥−2
d) 𝑓 𝑥 = 4 − 2𝑥 + 5
1
e) 𝑓 𝑥 = 𝑥 +1
f)
𝑒
−1
𝑓 𝑥 =2 𝑥−3 +4
g) 𝑓 𝑥 = ln
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𝑥+1
𝑥−3
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Representing functions
• Table
• Graph
• Formula
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Vertical line test
• Given a function 𝑓, every vertical line that may be drawn
intersects the graph of 𝑓 no more than once. If any
vertical line intersects a set of points more than once, the
set of points does not represent a function
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Zeros and 𝑦-intercepts of a function
• Zeros of a function : values of 𝑥 such that 𝑓 𝑥 = 0
• 𝑦-intercept of a function 𝑓: 𝑓 0
Example Consider the function 𝑓 𝑥 = 𝑥 + 3 + 1
a) Find all zeros of 𝑓
b) Find the 𝑦-intercept (if any)
c) Sketch a graph of 𝑓
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Increasing/decreasing function
• A function 𝑓 is increasing on the
interval 𝐼 if for all 𝑥1 , 𝑥2 ∈ 𝐼,
𝑓 𝑥1 ≤ 𝑓 𝑥2 when 𝑥1 < 𝑥2 .
• A function 𝑓 is decreasing on the
interval 𝐼 if for all 𝑥1 , 𝑥2 ∈ 𝐼,
𝑓 𝑥1 ≥ 𝑓 𝑥2 when 𝑥1 < 𝑥2 .
• A function 𝑓 is strictly increasing on
the interval 𝐼 if for all 𝑥1 , 𝑥2 ∈ 𝐼,
𝑓 𝑥1 < 𝑓 𝑥2 when 𝑥1 < 𝑥2 .
• A function 𝑓 is strictly decreasing
on the interval 𝐼 if for all 𝑥1 , 𝑥2 ∈ 𝐼,
𝑓 𝑥1 > 𝑓 𝑥2 when 𝑥1 < 𝑥2 .
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𝑓 𝑥 = 3𝑥
𝑓 𝑥 = −𝑥 3
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Increasing/decreasing function
• Example Determine which function is decreasing, increasing or
non-decreasing/increasing on its domain
• 𝑓 𝑥 = 𝑏 𝑥 , 0 < 𝑏 < 1, 𝐷 = ℝ
Consider 𝑥1 < 𝑥2 , then 𝑓 𝑥1 − 𝑓 𝑥2 = 𝑏 𝑥1 − 𝑏 𝑥2 = 𝑏 𝑥1 1 − 𝑏 𝑥2−𝑥1
𝑏 𝑥1 > 0
𝑥2 − 𝑥1 > 0
ൡ ⇒ 𝑓 𝑥1 − 𝑓 𝑥2 > 0 ⇒ 𝑓 decreases
ቅ ⇒ 𝑏 𝑥2−𝑥1 < 1
0<𝑏<1
• 𝑓 𝑥 = 𝑥 2𝑘+1 , 𝑘 ∈ ℕ
• 𝑓 𝑥 = cosh 𝑥
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𝑒 𝑥 +𝑒 −𝑥
=
,𝐷 =
2
0, +∞
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Combining functions
• Function composition: create a new function from
exist functions
𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥
Example 𝑓 𝑥 = 𝑥 2 , 𝑔 𝑥 = 3𝑥 + 1,
a) 𝑓 ∘ 𝑔 𝑥 = 𝑓 𝑔 𝑥 = 𝑓 3𝑥 + 1 = 3𝑥 + 1 2
b) 𝑔 ∘ 𝑓 𝑥 = 𝑔 𝑓 𝑥 = 𝑔 𝑥 2 = 3𝑥 2 + 1
• Combining functions with mathematical operators
a)
b)
𝑓±𝑔 𝑥 =𝑓 𝑥 +𝑔 𝑥
𝑓. 𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥
c)
𝑓
𝑔
𝑥 =
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𝑓 𝑥
𝑔 𝑥
for 𝑔 𝑥 ≠ 0
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Combining functions
• 𝑓: 𝐴 → 𝐵 , 𝑔: D → 𝐸 . If 𝐵 ⊆ 𝐷 , the composite
function 𝑔 ∘ 𝑓 𝑥 has domain A and
𝑔∘𝑓 𝑥 =𝑔 𝑓 𝑥
𝑔∘𝑓 1 =4
𝑔∘𝑓 2 =5
𝑔∘𝑓 3 =4
Example 𝑓 𝑥 = 𝑥 2 + 1, 𝑔 𝑥 = 1/𝑥
a) Find 𝑔 ∘ 𝑓 𝑥
b) Evaluate 𝑔 ∘ 𝑓 4 , 𝑔 ∘ 𝑓 −1/2
c) Find 𝑓 ∘ 𝑔 𝑥 and state its domain and range
d) Evaluate 𝑓 ∘ 𝑔 4 , 𝑓 ∘ 𝑔 −1/2
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Odd and even functions
• Odd function: 𝑓 −𝑥 = −𝑓 𝑥 , ∀𝑥 ∈ 𝐼 (symmetric about
the origin)
• Even function: 𝑓 −𝑥 = 𝑓 𝑥 , ∀𝑥 ∈ 𝐼 (symmetric about
the 𝑦-axis)
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Odd and even functions
• Example Determine whether each of the following
functions is even, odd, or neither.
a) 𝑓 𝑥 = −5𝑥 4 + 7𝑥 2 − 2
b) 𝑓 𝑥 = 2𝑥 5 − 4𝑥 + 5
3𝑥
c) 𝑓 𝑥 = 2
d)
e)
f)
g)
𝑥 +1
𝑓 𝑥
𝑓 𝑥
𝑓 𝑥
𝑓 𝑥
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= 𝑥
= 𝑥 𝑏 , with 𝑏 ∈ ℕ
= 𝑥 1/𝑛 , with 𝑛 ∈ ℕ, 𝑛 ≠ 0
=𝑒 𝑥 +1
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References
• E. Herman & G. Strang, Calculus 1 & 2, Opensta
https://openstax.org/details/books/calculus-volume-1
• W. K. Nicholson, Linear algebra with Application, LyryX Ver 2021-A
https://lyryx.com/linear-algebra-applications/
• https://assets.openstax.org/oscmsprodcms/media/documents/OpenStax.Effective.Reading.and.Noteta
king.Guide_E7WFZSP.pdf
• https://www.ocw.mit.edu/courses/18-01-calculus-i-single-variablecalculus-fall-2020/
• https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/
• Slides MAE101, FPT University, Quy Nhon
• Nguyễn Đình Trí, Tạ Văn Đĩnh, Nguyễn Hồ Quỳnh Bài tập Toán cao
cấp Tập 1 và Tập 2, NXB Giáo dục
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