Exercises - 1-1 and Inverse Functions

Here you will find problems on finding inverse functions, proving 1-1 using monotonicity and derivative, and finding inverse functions for more complicated functions. For interaction between elementary functions and their inverses see also Solving equations and inequalities in Functions - Theory - Sets and mappings.

If you want to refer to sections of Methods Survey while working the exercises, you can click here and it will appear in a separate full-size window. Similarly, here we offer Theory.

For each of the following functions, either find its inverse f₋₁ or show that f is not 1-1 on its domain.

• 1.
  Prove by definition that this function is monotone (which implies that it is 1-1).
  
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• 2.
  Prove by definition that this function is monotone (which implies that it is 1-1).
  
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• 3.
  Prove by definition that this function is monotone (which implies that it is 1-1).
  
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• 4.
  
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• 5.
  
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• 14.
  
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For each of the following functions, show that it is 1-1 on the indicated set M and find the inverse function to f restricted to M.
Hint for functions that are T-periodic: Find an integer k so that the shift x₀ = x − kT transforms M into an interval where the given function has its canonical inverse. Then solve for x₀ and pass to x. Sometimes a shift by T/2 is needed, a picture should help. Compare with note on inverse in Functions - Theory - Elementary functions - Trigonometric functions.

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• 20.
  
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