CK12 - Trigonometry

Example2:Statethe domainof eachfunction:

c. (2,4),(3,9),(5,11)

b.

a.y=x

2

Solution:

a.y=x^2

The domainof this functionis the set of all real numbers.Thereare no restrictions.

b.

The domainof this functionis the set of all real numbers,except. The domainis restrictedthis way
becausea fractionwith denominatorzero is undefined.

c. (2,4),(3,9),(5,11)

The domainof this functionis the set ofxvalues{2,3,5}

The variablexis oftenreferredto as theindependentvariable,whilethe variableyis referredto as the
dependentvariable.We talk aboutxandythis way becausetheyvaluesof a functiondependon whatthe
xvaluesare. That is why we also say that “yis a functionofx.” For example,the valueofyin the function
y= 3xdependson whatxvaluewe are considering.Ifx= 4, we can easilydeterminethaty= 3(4) = 12.
Returningto the situationin the introduction,we can say that the amountof moneyyou take independson
the numberof candybars you havesold.

Whenwe are workingwith a functionin the form of an equation,thereis a specialnotationwe can use to
emphasizethe fact thatyis a functionofx. For example,the equationy= 3xcan also be writtenasf(x) =
3 x. It is importantto rememberthatf(x) representstheyvalues,or the functionvalues,and that the letterf
isnota variable.That is,f(x) doesnotmeanthat we are multiplyinga numberfby anothernumberx. I like
to think of a functionas a machinethat takesin a number,x, and producesanothernumber. In the expression
f(x),fis the machineand the parenthesis( )are the placewherethe input,x, is enteredinto the machine.
f(x) is the outputthat the machineproduceswith the inputx. For example,considerthat your machineadds
5 to an input.Thenf(3) = 8, or moregenerally,f(x)=x+ 5.

Now that we haveconsideredthe domainof a function,we will turn to therangeof a function,whichis the
set of allyvaluesfor whicha functionis defined.Just as we did with domain,we can examinea function
and determineits range.Again,it is oftenhelpfulto thinkaboutwhatrestrictionstheremightbe, and what

the graphof the functionlookslike. Considerfor examplethe functiony=x^2. The domainof this function
is all real numbers,but whataboutthe range?

The rangeof the functionis the set of all real numbersgreaterthan or equalto zero.This is the case because
everyyvalueis the squareof anxvalue.If we squarepositiveand negativenumbers,the resultwill always

be positive.Ifx= 0, theny= 0. We can also see the rangeif we look at a graphofy=x^2 :