fn(x) = T x2 = 1+x + 2 + 6 k=0 n! + +

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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why can we conclude that f9: R → R has an inverse function? Does f10 have an inverse function?

n
fn(x) = E
x²
= 1+ x +
2
k!
k=0
n!
Transcribed Image Text:n fn(x) = E x² = 1+ x + 2 k! k=0 n!
Expert Solution
Solution:

Let us prove the equation by induction. Then, will determine whether inverse for the function exists or not.

By induction, we will have  f2n (x) is a positive number for all real numbers x.

Substitute n = 2n in the given equation.

f2n(x)=2nk=0xkk!=1+x+x22+x36+...+x2n2n!

Check the result for n = 1.

f2(x)=2k=0xkk!=1+x+x22

First and last terms are always positive.

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