The material in this section provides a basic review of and practice problems for pre-calculus skills essential to your success in Calculus. You should take time to review this section and work the suggested problems (checking your answers against those in the back of the book). Since this content is a pre-requisite for Calculus, reviewing and mastering these skills are considered your responsibility. This means that minimal, and in some cases no, class time will be devoted to this section. When you identify areas that you need help with we strongly urge you to seek assistance outside of class from your instructor or other student tutoring service.
We will briefly summarize the laws of exponents and equivalent forms of exponent expressions commonly used in this chapter. The laws of exponents are only valid for the values of x and y for which the expression is defined (i.e., nonzero real numbers in the denominator and nonnegative real numbers when roots are even.) Our first is the product of exponents. If and are real numbers, then
Our next is the quotient of exponents. If and are real numbers, then
Our third is when a power is raised to a power. Once again, we assume and are real numbers. In that case,
Our final law tells us how to distribute a power over a product and a quotient. If is a real number, then
The following examples demonstrate an efficient factoring technique that can be used to create the various equivalent expressions often needed to complete problems that arise in Calculus. The ability to move flexibly and efficiently among different representations of an expression is an important skill to have.
Factor completely to write an equivalent expression:
Factor out the lowest power of the common factor to simplify the complex fraction
Function composition refers to combining functions in a way that the output from one function becomes the input for the next function. In other words, the range (-values) of one function become the domain (-values) of the next function. We denote this as , where the output of becomes the input of .
Given and , find and .
SolutionTo find , we substitute the function into the function . Thus,
For , we substitute the function into the function . Thus,
Given , and , find and .
SolutionTo find we must start with the inside and work our way out.
For , we have
In this chapter we will also need to decompose a given function into two or more, less complex functions. For any one function there is often more than one way to write the decomposition. The following examples demonstrate this.
Given , find and so that .
SolutionOne solution is and .
Another possible solution is and .
In Exercises 1–4., simplify each expression. Write your answer so that all exponents are positive.
In Exercises 5–8., factor to write equivalent expressions.