I i TIC GEOMETRY (Cn. \. 



4. Identity, equation, and root. If two functions involv- 

 ing the same variables are equal to each other for all values 

 of those variables they are identically equal. Such an 

 equality is expressed by writing the sign = between tin 

 two functions, and the expression so formal is an identity. 

 If, nn tin- nther hamL tin- t\\" functions are equal to each 

 Other only f>r part icular values of the variables, tin- equality 

 is expressed by writing the sign = between the two filia- 

 tions, and the expression so formed in an equation. 11 1 

 particular values for \vhirh the two functions an equal, 

 those values of the variables which satisfy the equation, are 

 the roots nf the equation. 



**. (* + *)'=** + - f u + f> ( T + ")(* - fl ) + fl '=* a 



and , + a i= 



x-\ x-l 



are identities; while 3x* - 10x +2 = 2z 2 - 4x - 0, or, what is the same 

 thing, x* - 6z + 8 = 0, is an equation. The roots of this equation are 

 the numbers 2 and 4. 



Special attention is called to the fact that an equation 

 always imposes a condition. 



7., x* 6z + 8 = if, and only if, x = 2 or x 4. So also the equa- 

 tion ax + 6jf + c = imposes the condition that x shall be equal to 



5. Functions classified. A functional relation is usually 

 r-essed by means of an conation involving the related 

 numbers. If the form of this equation is such that one of 

 the variables is expressed directly in terms of the other-, then 

 that variable is called an explicit function of the others: if 

 it is not so expressed, it is an implicit function. 



Bj., the equations y = V5-x x* 4- y = 5, and x = \/5 - y* express 

 the MOM relation between x and y ; in the first y is an explicit function 



