8/23/2024
Thomas’ Calculus: Early Transcendentals
Fifteenth Edition
Chapter 1
Functions
Slide - 1
Section 1.2
Combining Functions;
Shifting and Scaling Graphs
Examples Exercises
1, 2, 3, 4(c) 2, 4, 18
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In this section we look at the main ways functions are
combined or transformed to form new functions.
± ,× ,÷ , °
reflecting
Shifting
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Sums, Differences, Products, and Quotients
Two functions 𝑓 and 𝑔 can be combined to form new functions 𝑓 ± 𝑔, 𝑓𝑔
and 𝑓/𝑔 , 𝑓 ∘ 𝑔 as follows:
1 𝑓 ± 𝑔) (𝑥 = 𝑓 𝑥 ± 𝑔(𝑥) remark
If 𝐷 = 𝐴 , 𝐷 = 𝐵 then:
𝑓 𝑔
2 𝑓 ⋅ 𝑔 𝑥 = 𝑓 𝑥 ⋅ 𝑔(𝑥)
1 𝐷𝑓±𝑔 = 𝐴 ∩ 𝐵
3
𝑓
𝑥 =
𝑓 𝑥 2 𝐷𝑓⋅𝑔 = 𝐴 ∩ 𝐵
𝑔 𝑔(𝑥)
3 𝐷𝑓∕𝑔 = 𝐴 ∩ 𝐵 − {𝑥: 𝑔 𝑥 = 0}
4 𝑓 ∘ 𝑔 𝑥 = (𝑓(𝑔 𝑥 )
4 𝐷𝑓 ∘𝑔 = 𝐷𝑓∘𝑔 ∩ 𝐷𝑔
5 𝐷𝑔 ∘𝑓 = 𝐷𝑔∘𝑓 ∩ 𝐷𝑓
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Example 1 c 𝑔 𝑥 −𝑓 𝑥
Let 𝑓 𝑥 = 𝑥 , 𝑔 𝑥 = 1 − 𝑥.
Find each function and its domain:
d 𝑓(𝑥). 𝑔 𝑥 =
a 𝑓 𝑥 +𝑔 𝑥
e 𝑓 𝑥
𝑔 𝑥
b 𝑓 𝑥 −𝑔 𝑥
𝑔 𝑥
f
𝑓 𝑥
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Exercise 2
Find the domains of 𝑓 , 𝑔 , 𝑓 + 𝑔 and 𝑓. 𝑔 .
𝑓 𝑥 = 𝑥+1 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 − 1
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Exercise 4
𝑓 𝑔
Find the domains of 𝑓 , 𝑔 , and .
𝑔 𝑓
𝑓 𝑥 =1 𝑎𝑛𝑑 𝑔 𝑥 = 1 + 𝑥
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Composite Functions
Definition
If 𝑓 and 𝑔 are functions, the composite function 𝒇 ∘ 𝒈 (“f composed
with g”) is defined by
𝑓 ∘ 𝑔 𝑥 = 𝑓(𝑔 𝑥 )
The domain of 𝒇 ∘ 𝒈 consists of the numbers
𝑥 in the domain of g for which 𝑔(𝑥) lies in
the domain of 𝑓.
𝐷𝑓 ∘𝑔 = 𝐷𝑓∘𝑔 ∩ 𝐷𝑔
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Composite Functions
A composite function f g uses the output g ( x)
of the first function g as the input for the second function f.
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Example 2
If 𝑓 𝑥 = 𝑥 , 𝑔 𝑥 = 𝑥 + 1 . Write formula for
𝑎 𝑓∘𝑔 𝑥 𝑏 𝑔𝑜𝑓 𝑥 𝑐 𝑓∘𝑓 𝑥 𝑑 (𝑔 ∘ 𝑔)(𝑥)
and find the domain of each.
𝐷𝑓 ∘𝑔 = 𝐷𝑓∘𝑔 ∩ 𝐷𝑔
𝐷𝑔 ∘𝑓 = 𝐷𝑔∘𝑓 ∩ 𝐷𝑓
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Exercise 18
(a) write formulas for 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓
(b) find the domain of each.
𝑓 𝑥 = 𝑥2 , 𝑔 𝑥 = 1 − 𝑥
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Shifting and Scaling Graphs
reflecting
Shifting
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Vertical and Horizontal Shifts
Suppose 𝑘 > 0. To obtain the graph of
Vertical Shifts
𝑦 = 𝑓(𝑥) + 𝑐, shift the graph of 𝑦 = 𝑓(𝑥) a distance 𝑘 units upward.
𝑦 = 𝑓 𝑥 − 𝑐, shift the graph of 𝑦 = 𝑓(𝑥) a distance 𝑘 units downward.
Horizontal Shifts
𝑦 = 𝑓(𝑥 − 𝑐), shift the graph of 𝑦 = 𝑓(𝑥) a distance 𝑘 units to the right.
𝑦 = 𝑓(𝑥 + 𝑐), shift the graph of 𝑦 = 𝑓(𝑥) a distance 𝑘 units the left.
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Example 3
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Vertical and Horizontal Reflecting Formulas
For c = −1, the graph is reflected:
y = − f ( x) Reflects the graph of f across the 𝑥 − 𝑎𝑥𝑖𝑠.
y = f (− x) Reflects the graph of f across the 𝑦 − 𝑎𝑥𝑖𝑠.
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Example 4 (c)
Let 𝑓 𝑥 = 𝑥. Sketch the graph
𝑔 𝑥 =− 𝑥+1 𝑎𝑛𝑑 ℎ 𝑥 = −𝑥 + 1
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TYPES OF TRANSFORMATIONS
OUTSIDE INTSIDE
𝒚- VALUES OR VERTICAL 𝒙- VALUES OR HORIZONTAL
𝑓 𝑥 + 𝑘 𝑠ℎ𝑖𝑓𝑡 𝑢𝑝 𝑓 𝑥+𝑘 𝑠ℎ𝑖𝑓𝑡 𝑙𝑒𝑓𝑡
𝑓 𝑥 − 𝑘 𝑠ℎ𝑖𝑓𝑡 𝑑𝑜𝑤𝑛 𝑓 𝑥−𝑘 𝑠ℎ𝑖𝑓𝑡 𝑟𝑖𝑔ℎ𝑡
−𝑓 𝑥 𝑟𝑒𝑓𝑙𝑒𝑐𝑡 𝑜𝑣𝑒𝑟 𝑥 − 𝑎𝑥𝑖𝑠 𝑓 −𝑥 𝑟𝑒𝑓𝑙𝑒𝑐𝑡 𝑜𝑣𝑒𝑟 𝑦 − 𝑎𝑥𝑖𝑠
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HOMEWORK
1,3,5(a,c,g),23,25,57 (a,b,f)
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