Solved Problems for Functions: Basic Properties

Here we will cover the most typical approaches to solving these problems. The solutions are usually straightforward, directly using the definition.
Note that solved problems concerning monotonicity and concavity solved using derivatives (which is the recommended way) can be found in Derivative - Solved Problems - Graphing functions. Here we just show a couple of monotonicity problems solved by definition.

If you wish to simultaneously follow another text on basic properties of real functions in a separate window, click here for Theory and here for Methods Survey.

• Problem: Determine the domain of the function

  

• Problem: Determine the domain of the function

  

Note: The domain is also investigated in the next four problems. We also find domain in most problems in Derivatives - Solved Problems, of interest are for instance this, this, this, this, this, and this.

• Problem: Investigate boundedness of the function

  

• Problem: Determine symmetry of the function

  

• Problem: Determine symmetry of the function

  

Note: Symmetry is also investigated in problems on graph sketching in Derivatives - Solved Problems, of interest are for instance this, this, and this.

• Problem: Determine periodicity of the function

  

Note: Periodicity plays important role in this problem in Derivative - Solved problems.

• Problem: Determine whether the following function is 1-1, and if the answer is positive, find the corresponding inverse function. Also investigate its monotonicity.

  

• Problem: Determine whether the following function is 1-1, and if the answer is positive, find the corresponding inverse function. Also investigate its monotonicity.

  

• Problem: Determine whether the following function is 1-1, and if the answer is positive, find the corresponding inverse function. Also investigate its monotonicity.

  

Note: Monotonicity is investigated in the recommended way in Derivatives - Solved Problems - Graphing functions, of interest are for instance this, this, this, and this.

• Problem: Investigate continuity of the function

  

• Problem: Determine constants a,b for which the following function is continuous:

  

• Problem: Using the knowledge of transformations, guess the graphs of the function

  

• Problem: Using the knowledge of transformations, guess the graphs of the function

  

• Problem: Using the knowledge of transformations, guess the graphs of the function

  

  (Compare the role of the order of transformations)