> <\body> |>>> <\definition> A function is a to assign to a variable, say , an value, say . In general, we can write <\equation*> y=f. Here <\itemize> is the name of the function is called the is called the The set of values that can take is called the of the function. The set of all possible values of constitute the . > is read We can view a function as a machine, that takes the input , and gives the output . The of a > is the curve <\equation*> \\:y=f|}>. >: the plane |gr-frame|>|gr-geometry||gr-color|red|||>>|||>>||||>>||>>|>>|>>||>>|>>|>||>>||>>|>>| This represents the\ graph of a function >>||>||>||||>>||>>|>|>|>|>>||>>||>>||>>| This curve doesn't represent the graph of a function >>>> <\em> <\equation*> y=|0>>||0>>>>> This function can also be written as <\equation*> y=. (even/odd functions) functions <\itemize> linear functions <\equation*> y=m*x+b where is the , and is the -. For a linear function, if =m*x+b,y=m*x+b>, then <\equation*> y-y=m*-x|)>. Graphing calculator: www.desmos.com power functions <\equation*> y=x The property depends on the value of . <\itemize> If >, then it's an even function, and it's increasing function for 0> and decreasing for 0>. If >, then it's an odd function, and it's increasing function for all . If >, then it's an even function, and it's decreasing function for 0> and increasing for 0>. If >, then it's an odd function, and it's decreasing function for 0> or 0>. If ,,,\>, then the function is defined only for 0>, and it's an increasing function. If ,,,\>, then the function is defined for all , and it's an increasing function. other cases: read the slides or book polynomials <\equation*> y=a*x+a*x+\+a*x+a, where >is called the of the polynomial, and >'s are called the . For example, -x+5> is a polynomial of degree 2, or we call it a , and +2x-10> is a polynomial of degree 3, also called a . The domain of any polynomial is =,\|)>>. rational functions <\equation*> y=|Q>, where are both polynomials. For example <\equation*> y=-2x|4x-5> The domain of a rational function is all except where =0>. algebraic functions: any function the can be obtained by addition, subtraction, multiplication, and raising to rational powers of . For example <\equation*> y=+3*|x-4>+10*x. exponential functions <\equation*> y=b, where 0> is called the , is called the . logarithmic functions <\equation*> y=log*x they are actually the of exponential functions, i.e. <\equation*> y=log x\x=b trigonometric functions <\equation*> y=sin x,cos x,tan x,cot x,sec x,csc x Elementary building blocks:\ <\enumerate> power functions exponential functions logarithmic functions trigonometric functions <\itemize> translations (move the graph up/down, left/right) <\itemize> vertical translation <\equation*> y=f+c horizontal translation <\equation*> y=f stretching and reflecting <\itemize> vertical stretching/reflecting <\equation*> y=c*f horizontal stretching/reflecting <\equation*> y=f|)> algebraic combinations <\equation*> f+g,f-g,f*g, composition <\equation*> f|)>=g> read as of of , or composed with <\example> If =x+3*x> and =sin x>. Then <\itemize> g=f>|)>=>|)>+3*>=+3*sin x> f=g|)>=sin f=sin +3*x|)>.> f=|)>+3*f=+3*x|)>+3*+3*x|)>> g=sin g=sin .> <\equation*> x|g>g|gr-arrow-end|\||||>>|g|>>>|2cm>|f>f|)> <\example> Write the function =log +1>> as a composition of 3 functions. Answer: Let =log x,g=,h=x+1>. Then <\equation*> F=f\g\h Application example (compound interest): Suppose you deposit an amount of > in an account with annual interest rate of . The balance after years, if the interest is compounded <\itemize> annualy, then <\equation*> u=u* monthly, then <\equation*> u=u*|)>=u*|)>|]> times per year, then <\equation*> u=u|)> continuously, then <\equation*> u=lim\>u|)>=lim\>u|)>|]>=u*e, where > is called the . Here we are using the formula <\equation*> \>|)>=e> \; \; \; \; \; \; \; > <\initial> <\collection> <\references> <\collection> > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||1Functions and Models> |.>>>>|> |math-font-series||2A catalog of essential functions> |.>>>>|> |math-font-series||3New functions from old> |.>>>>|>