We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.
Linear Functions and SlopeThe easiest type of function to consider is a linear function. Linear functions have the form \(f(x)=ax+b\), where \(a\) and \(b\) are constants. In Figure \(\PageIndex{1}\), we see examples of linear functions when a is positive, negative, and zero. Note that if \(a>0\), the graph of the line rises as \(x\) increases. In other words, \(f(x)=ax+b\) is increasing on \((−∞, ∞)\). If \(a0\), the values \(f(x)→∞\) as \(x→±∞\). If \(a0\).; the parabola opens downward if \(a0\), then \(f(x)→∞\) as \(x→∞\) and \(f(x)→−∞\) as \(x→−∞\). If \(a0\),the parabola opens upward. If \(a0\), the values \(f(x)→∞\) as \(x→∞\) and the values \(f(x)→−∞\) as \(x→−∞\). If the leading coefficient \(a0\), Equation \ref{quad} tells us there are two real numbers that satisfy the quadratic equation. If \(b^2−4ac=0\), this formula tells us there is only one solution, and it is a real number. If \(b^2−4ac0,\, b≠1\). A logarithmic function is a function of the form \(f(x)=\log_b(x)\) for some constant \(b>0,\,b≠1,\) where \(\log_b(x)=y\) if and only if \(b^y=x\). (We also discuss exponential and logarithmic functions later in the chapter.)
Example \(\PageIndex{7}\): Classifying Algebraic and Transcendental FunctionsClassify each of the following functions, a. through c., as algebraic or transcendental.
\(f(x)=\dfrac{\sqrt{x^3+1}}{4x+2}\)\(f(x)=2^{x^2}\)\( f(x)=\sin(2x)\)SolutionSince this function involves basic algebraic operations only, it is an algebraic function.This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)As in part b, this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.Exercise \(\PageIndex{5}\):Is \(f(x)=x/2\) an algebraic or a transcendental function?
AnswerAlgebraic
Piecewise-Defined FunctionsSometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function. The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of \(x\):
\[f(x)=\begin{cases}−x, & \text{if } x