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General Mathematics
Composition of Functions
For all values of x for which both f(ξ΅) and g(ξ΅) are defined, we define the following as:
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅σ°ξ΅
σ°
ξ΅
σ°
σ°
In general, the notation ξ΅σ°ξ΅
σ°
ξ΅
σ°
σ°, read as βξ΅ of ξ΅ of ξ΅,β means βξ΅ composed with ξ΅ of ξ΅β can be
written as
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
.
Illustrative Examples:
Given the functions ξ΅
σ°
ξ΅
σ°
ο ξ΅
οΆ
ο ξ₯³ and ξ΅
σ°
ξ΅
σ°
ο ξ΅ ο ξ₯³, find the following:
a.
σ°
ξΊ ξ― ξΊ
σ°σ°
ξΊ
σ°
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅ο«ξ΅
σ°
ξ΅
σ°
ο―
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅
σ°
ξ΅ ο ξ₯³
σ°
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο
σ°
ξ΅ ο ξ₯³
σ°
οΆ
ο ξ₯³
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅
οΆ
ο ξ₯΄ξ΅ ο ξ₯³ ο ξ₯³
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅
οΆ
ο ξ₯΄ξ΅
b.
σ°
ξΊ ξ― ξΊ
σ°σ°
ξΊ
σ°
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅ο«ξ΅
σ°
ξ΅
σ°
ο―
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅σ°ξ΅
οΆ
ο ξ₯³σ°
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο
σ°
ξ΅
οΆ
ο ξ₯³
σ°
ο ξ₯³
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅
οΆ
ο ξ₯³ ο ξ₯³
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅
οΆ
c.
σ°
ξΊ ξ― ξΊ
σ°σ°
ο
σ°
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅
οΆ
ο ξ₯΄ξ΅
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ₯²
σ°
ο ξ₯²
οΆ
ο ξ₯΄σ°ξ₯²σ°
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ₯²
σ°
ο ξ₯²
Alternative Solution:
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ₯²
σ°
ο ξ΅σ°ξ΅
σ°
ξ₯²
σ°
σ°
ξ΅
σ°
ξ₯²
σ°
ο ξ₯² ο ξ₯³
ξ΅
σ°
ξ₯²
σ°
ο ξ₯³
ξ΅ο«ξ΅
σ°
ξ₯²
σ°
ο― ο ξ΅
σ°
ξ₯³
σ°
ξ΅
σ°
ξ₯³
σ°
ο σ°ξ₯³σ°
οΆ
ο ξ₯³
ξ΅
σ°
ξ₯³
σ°
ο ξ₯²
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ₯²
σ°
ο ξ₯²
d.
σ°
ξΊ ξ― ξΊ
σ°σ°
οο
σ°
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°
ο ξ΅
οΆ
σ°
ξ΅ ξ― ξ΅
σ°σ°
οξ₯³
σ°
ο σ°οξ₯³σ°
οΆ
σ°
ξ΅ ξ― ξ΅
σ°
σ°οξ₯³σ° ο ξ₯³
e.
σ°
ξ΅ ξ― ξ΅
σ°σ°
ξ΅
σ°

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General Mathematics Composition of Functions For all values of x for which both f(π₯) and g(π₯) are defined, we define the following as: (π β π)(π₯) = π(π(π₯)) In general, the notation π(π(π₯)), read as βπ of π of π₯,β means βπ composed with π of π₯β can be written as (π β π)(π₯). Illustrative Examples: Given the functions π(π₯) = π₯ 2 β 1 and β(π₯) = π₯ + 1, find the following: a. (π β π)(π) (π β β)(π₯) = π(β(π₯)) (π β β)(π₯) = π(π₯ + 1) (π β β)(π₯) = (π₯ + 1)2 ? ...
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