Finding the inverse of two functions

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SUMMARY

To find the inverse of the functions y = e^(-x^3) and y = sin(1/x), specific steps must be followed. For y = e^(-x^3), the first logical step is to take the natural logarithm of both sides to isolate x. In contrast, y = sin(1/x) presents a challenge as it is not invertible due to its many-to-one nature, with infinitely many x values yielding the same y value. Understanding these properties is crucial for solving for x in terms of y.

PREREQUISITES
  • Understanding of exponential functions and their properties
  • Knowledge of logarithmic functions, specifically natural logarithms
  • Familiarity with trigonometric functions and their inverses
  • Concept of function invertibility and restrictions on domains
NEXT STEPS
  • Study the properties of exponential functions and their inverses
  • Learn about the natural logarithm and its applications in solving equations
  • Explore the characteristics of trigonometric functions and their restrictions
  • Investigate methods for determining the invertibility of functions
USEFUL FOR

Students studying calculus, mathematicians interested in function analysis, and educators teaching inverse functions and their properties.

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Homework Statement



How do I find the inverse of these functions step by step?

y= e^-x^3

y= sin(1/x)

I know the solutions but I don't know how to work with these two functions. Does anyone know the steps to finding the inverse of these?
 
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The goal is to solve for x in terms of y.

What can you do to both sides of y = e^(-x^3) that would be a logical first step?

For the second function, y = sin(1/x), is there a restriction on the domain? As it's written, this function is not invertible because it's many-to-one. For example, there are infinitely many x for which sin(1/x) = 0, namely x = 1/(n*pi) for any nonzero integer n.
 
Tebow15 said:

Homework Statement



How do I find the inverse of these functions step by step?

y= e^-x^3

y= sin(1/x)

I know the solutions but I don't know how to work with these two functions. Does anyone know the steps to finding the inverse of these?
How do you find the inverse of any function, in general?
 

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