Types of Functions Algebraic Functions

1 MATH 1170 Chapter 1 Worksheet #1 NAMENote:It Ibolded and underlineda term, you are responsible for a verbadom definition of that term(as well as understanding that definition). If I justboldeda term, I only expect that you are comfortablewith the use of and using that term ( understanding it). Types of FunctionsNow that we have discussed what Functions are and some of their characteristics, we will explore differenttypes of fumctions. Section of the text outlines a variety of Types of Functions . Notice that since thefollowing are all Functions , they will all pass the Vertical Line FunctionsA function is called analgebraic functionif it can be constructed using Algebraic operations (such asaddition, subtraction, multiplication, division and taking roots).

2 Polynomials, power Functions , and rational function are all Algebraic PolynomialsA functionpis apolynomialifp(x) =anxn+an 1xn 1+..+a2x2+a1x+a0wherenis a nonnegative integer anda0,a1,a2,..,an 1,anare all constants calledcoefficientsof the leading coefficientan6= 0 then thedegreeofp(x) isn. A polynomial of degree 1 is called alinear function. A polynomial of degree 2 is called aquadratic function. A polynomial of degree 3 is called acubic function. A polynomial of degree 4 is called aquartic function. A polynomial of degree 5 is called aquintic function.

3 And so often talk about findingrootsof a polynomial. This means we are finding where the graph of thefunction hits thex-axis. Finding the roots of a polynomial entails setting the polynomial equal to zero andsolving the roots ofp(x) =x2 x FunctionsThe most famous polynomial is the linear said to be alinear functionofxif the graph of the function is a line so that we can use theslope-intercept formof the equation of a line to write a formula for the function asy=mx+bwheremis the slope andbis this is the case, what domandbequal in thep(x) equation?

4 Graph the family of equationsf(x) =x+bwhere be is an integerb= 2, 1,0,1,2 on the same the family of equationsf(x) =mxwhere be is an integerm= 2, 1,0,1,2 on the same the equation for a line that passes through (2, 1) and (3,5).2 Power FunctionsA function of the formf(x) =xawhereais a constant is called apower function. The power functiontakes a variety of forms based on the type of constant different forms arrise when isna positive integer, 1/nwherenis a positive integer, and will explore these forms in the following Ifa=nwherenis a positive that isnis a positive integer then the power function is really just a type your graphing calculator, sketch a graph of the following (x) = (x) = (x) = (x) = (x) = (x) =x6 Whennisodd, what is the domain off(x) =xn?

5 What is the range off(x) =xn? where isf(x) =xnincreasing? where isf(x) =xndecreasing? How do you know that these characteristics will hold foreveryoddn?Whenniseven, what is the domain off(x) =xn? what is the range off(x) =xn? where isf(x) =xnincreasing? where isf(x) =xndecreasing? How do you know that these characteristics will hold foreveryevenn? Ifa= 1/nwherenis a positive Functions of the formf(x) =x1/nare calledroot is important to note thatroot Functions are the inverses of polynomial Functions . But what does thismean?Recall the definition of an inverse function from one of our previous lectures.

6 (It can also be found on page62 of the Stewart text.)What does it mean to say that root Functions are the inverses of polynomial Functions ?Demonstrate thatg(x) =x1/2is the inverse function off(x) = (x) =x1/nin a different (x) =x1/n, wherenis a positive integer, a polynomial?Using your graphing calculator as a tool, sketch a graph of the following (x) =x1 (x) =x1 (x) =x1 (x) =x1/5 Whennisodd, what is the domain off(x) =x1/n? what is the range off(x) =x1/n? where isf(x) =x1/nincreasing? where isf(x) =x1/ndecreasing? How do you know that these characteristics will hold foreveryoddn?

7 4 Whenniseven, what is the domain off(x) =x1/n? what is the range off(x) =x1/n? where isf(x) =x1/nincreasing? where isf(x) =x1/ndecreasing? How do you know that these characteristics will hold foreveryevenn? Ifa= this case,f(x) =x 1is thereciprocal (x) =x 1a polynomial?Using your graphing calculator as a tool, sketch a graph off(x) =x 1and describe the domain, rangeand intervals of increasing and decreasing:Domain:Range:Increasing:Decre asing:3 Rational FunctionsMoving on from power Functions , we will now explore our last type of Algebraic function: the functionis a ratio of two polynomials,p(x) andq(x):f(x) =p(x)q(x)Which of the previously mentioned Functions is a rational function?

8 What happens when you evaluate thisfunction atx= 0?5 Considerf(x) =x 1x2 4. Identifyp(x) andq(x). Using your graphing calculator, sketch a graph off(x). What happens atx= 2 andx= 2? Why? exponential FunctionsTheexponential functionsare the Functions of the formf(x) =ax, where the baseais a that these function are called exponential Functions because the variable,x, is in the your graphing calculator as a tool, sketch a graph of the following Functions and describe the domain,range and intervals of increasing and (x) = (x) = (1/2) (x) = (x) = (1/3)xWhena >1, what is the domain off(x) =ax?

9 What is the range off(x) =ax? where isf(x) =axincreasing? where isf(x) =axdecreasing? How do you know that these characteristics will hold foreverya >1?6 When 0< a <1, what is the domain off(x) =ax? what is the range off(x) =ax? where isf(x) =axincreasing? where isf(x) =axdecreasing? How do you know that these characteristics will hold forevery0< a <1?What is the difference between the functionf(x) =x2andg(x) = 2x?Find the exponential functionf(x) =axwhose graph goes through the point ( 4,1/16).Logarithmic FunctionsThelogarithmic Functions ,f(x) = logax, where the baseais a positive constant, are the Functions thatare the inverse of the exponential is important to realize thatlogarithmic Functions are the inverses of exponential does it mean to say thatf(x) = logaxis the inverse ofg(x) =ax?

10 Show thatf(x) = log2xis the inverse ofg(x) = 2x. (Hint:This may require you to use the laws oflogarithms that can be found on page 65 on Stewart.)7 Using your graphing calculator as a tool, sketch a graph of the following Functions and describe the domain,range and intervals of increasing and (x) = (x) = log10xDomain:Domain:Range:Range:Increasi ng:Increasing:Decreasing:Decreasing:Trig onometric FunctionsUsing your graphing calculator as a tool, sketch a graph of the following Functions and describe the domain,range and intervals of increasing and (x) = (x) = (x) = tanxDomain:Domain:Domain:Range:Range:Ran ge:Increasing:Increasing:Increasing:Decr easing:Decreasing.