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Lecture 1

# MTH 162 Lecture 1: 5.1 Inverse Functions Notes Premium

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School
University of Miami
Department
Mathematics
Course
MTH 162
Professor
Pachero
Semester
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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