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Lecture 1

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University of Miami

Mathematics

MTH 162

Pachero

Fall

Description

MTH 162
Calculus II
5.1 Inverse Functions Notes
L. Sterling
Abstract
Provide a generalization to each of the key terms listed in this section.
Inverse Function
What is an inverse function?
▯ The given correspondence from f’s range back to f’s domain, which can also be described as
f▯1.
Domain and Range
f’s Domain=f ▯1’s Range
f’ :Range=f▯1 ’s Domain
▯1
f and f
x ! f (x) ! f▯1(f (x)) = x
Domain of f : f▯1(f (x)) = x
▯1 ▯ ▯1 ▯
x ! f (x) ! f f (x) = x
▯1 ▯ ▯1 ▯
Domain of f : f f (x) = x
Graph Theorem
▯ What is the graph theorem?
▯1
– A function’s graph, f, and f , which is its inverse, are fully symmetric, but with
respect to the line, which is y = x.
Finding Inverse Functions
▯ What are the steps into ﬁnding an inverse function?
– Since y = f(x), interchange both x and y, which are the variables, to create x = f(y),
▯1
which implies f , which would be the inverse function, implicitly.
– Solve the implicit equation for y, which would be in terms of x in order to ’stain f
explicit form (if possible). ▯ ▯
– Check your result(s) by showing that both f(f (x)) = x and f ▯1(x) = x.
1 Inverse Function Example
f(x) = x + 3
y = x + 3
x = y + 3
x ▯ 3 = y + 3 ▯ 3
x ▯ 3 = y
y = x ▯ 3
▯1
f (x) = x ▯ 3
Rational Inverse Function Example
1
f (x) = x3
1
y = x3
x = 13
▯ 3▯ 1 ▯ 3▯
y x = y3 y
3
y x = 1
y x 1
x = x
y = 1
p q
3 y = 3 1
q x
3 1
y = x
q
▯1 3 1
f (x) = x
Radical Inverse Function Example
p
f (x) = x +

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