when a system of definite values is given to the others, it is called a FUNCTION of those others. The function itself is a dependent variable, and the variables to which values are given are independent variables. The usual notation to express that one variable y is a function of another x is y=f(x), or y=F(x), or y= p(x); the letters f), F( ), $( ), x( ), . . . being generally retained to represent functions of arbitrary or unknown form. If u be an arbitrary or unknown function of several variables x, y, z, we may express the fact by the equation u = f(x, y, z). Ex. In any triangle, two of whose sides are x and y and the included angle 0, we have A=xy sin 0 to express the area. Here is the dependent variable, and is a function of known form—of x, y, and 8, which are the independent variables. 7. It will be seen that we could write the same equation in other forms, which may be regarded as an expression for sin in terms of the area and two sides; so that now sin may be regarded as the dependent variable, while ▲, x, y, are independent variables. And it is clear that if there be one equation between four variables, as above, it is sufficient to determine one in terms of the other three, so that any one variable may be regarded as dependent and the others as independent. This may be extended. For, if there be one equation between n variables, it will suffice to find one of them in terms of the remaining (n − 1), so that any one variable can be considered dependent and the remaining (n−1) independent. And, further, if there be equations connecting n variables (n being greater than r) they will be enough to determiner of the variables in terms of the other n— variables, so that any r of the variables can be considered dependent, while the remaining (n—r) are independent. 8. Explicit and Implicit Functions. A function is said to be EXPLICIT when expressed directly in terms of the independent variable or variables. or For example, if z=x2, or z=r sin 0, or z= x2y, z=ave log x+(a+x)" : 2 is expressed directly in terms of the independent variables, and is therefore in each of the above cases said to be an explicit function of those variables. But, if the function be not expressed directly in terms of the independent variable (or variables), the function is said to be IMPLICIT. If, for example, or or or ax2+yx − b=0); x+y=3axy; ax2+2bxy+cy2+2dx + 2ey+f=0; y in each case is said to be an implicit function of x. Sometimes, however, we can solve the equation for y: e.g., the It appears then that if the equation connecting the variables be solved for the dependent variable, that variable is reduced from being an implicit to being an explicit function of the remaining variable or variables. Such solution is not, however, always possible or convenient. 9. Species of Known Functions. Functions which are made up of powers of variables and constants connected by the signs + Xare classed as algebraic functions. If radical signs or fractional indices occur in the function, it is said to be irrational; if not, rational. All other functions are classed as transcendental functions. Of transcendental functions, sines, cosines, tangents, etc., are called trigonometrical or circular functions. Functions such as sin, tan-lx, etc., are called inverse trigonometrical functions. Functions such as e, aa2, in which the variable occurs in the index, are called exponential functions. While if logarithms are involved, as for instance in logex or log1(a+bx), etc., the function is called log arithmic. Besides the above we have the hyperbolic functions, sinh x, cosh x, etc., of which a short description follows in Art. 25. 10. Limit of a function. DEF. When a function can be made to approach continually to equality with some fixed value or condition so as to differ from it by less than any assignable quantity, however small, by making the independent variable or variables approach some assigned value or values, that fixed value or condition is called the LIMIT of the function for the value or values of the variable or variables referred to. 11. Illustrations. Ex. 1. If an equilateral polygon be inscribed in any closed curve, and the sides of the polygon be decreased indefinitely and at the same time increased in number indefinitely, the polygon continually approximates to the form of the curve, and ultimately differs from it in area by less than any assignable magnitude, and the curve is said to be the limit of the polygon inscribed in it. 3 is 2x+3 and x+1 X can be x+1 ; and by diminishing a indefinitely x+1 made less than any assignable quantity however small. that if x be increased indefinitely it can be made to continually approach and to differ by less than any assignable quantity from 2, which is therefore its limit in that case. Ex. 3. The limits of some quantities are zero, e.g., ax2+ bx, sin x, 1- cos x, 1- sin x, cos x, when x is zero, } When the limit of a quantity is zero for any value or values of the independent variable or variables, the quantity is said to be a vanishing quantity for those values. It is useful to adopt the notation Ltz-a to denote the words "the limit when x=a of." Ex. 4. The sum of a G.P. of which the first term is a, Jon 1 common ratio r, and n the number of terms, is a- 7' 1 If r < 1, the sum to infinity is a - For the differ ; and since Ltn=α =0 (when < 1), this -1 difference is a vanishing quantity. Ex. 5. We say '6, by which we mean that by taking enough sixes we can make 666... differ by as little as we please from 3. Ex. 6. The DEFINITION OF A TANGENT is another example. DEF. Let PQ be a chord joining P, Q, two adjacent points on a curve. Let Q travel along the curve towards P and come so close as ultimately to coincide with P. Then the limiting position of PQ, viz. PT, is called the tangent at P. T P Fig. 1. The angle QPT is a vanishing quantity; for it can be made less than any assignable quantity by making Q move along the curve sufficiently close to P. |