Solving Problems Using Properties of Functions

Abstract

Many students struggle to understand functions and their behavior, especially such concepts as the domain and range of a function. We very often start manipulations with an equation without thinking first about the properties of the functions. For example, the solution to the equation, $\sqrt{x}+\sqrt{x+16}=8$, can be found immediately as x = 9 if, instead of the standard squaring of both sides technique, we notice that the left side of the equation is the sum of two monotonically increasing functions. This means that if a solution exists, it must be unique. We can see that x = 9 makes the equation true; therefore, it is the only solution.

There are many different “tricks” and properties that you will learn in this chapter, but the chapter is mainly focused on recognizing boundedness of functions and using this property in the solution of equations. Likewise, the solutions of the transcendental equation ${ \sin}^2\left(\pi x\right)+\sqrt{x^2+3x+2}=0$ can be found if we notice that the left side is the sum of two nonnegative functions while the right side is zero. In this chapter you will learn that the solution exists only if each function on the left is zero. The purpose of this chapter is to help you understand these topics at the introductory level and demonstrate how knowledge of the properties of functions allows us to solve nonstandard problems of elementary mathematics.