mathematics

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Sums, differences, products, and quotients of functions
Like numbers, functions can be added, subtract, multiplied and divided to produce new functions.
If f and g are functions then:
(f+g) (x) = f(x)+g(x)
(f-g)(x)= f(x) –g(x)
(f.g) (x)=f(x).g (x)
(f/g) (x) =f (x) /g (x) where g (x) doesn t equal zero.
(c f) (x) = c f (x), c= constant
Composite functions

If f(x)= , g(x)=x+1 then

(g )(x)=g( )
(f f )(x)= f(f(x))=x1/4

H.W: find f+g and f.g and their domain and ranges:

1. f(x)=x, g(x)=

2. F (x) =x+5, g (x) =x2-3 find: F (g (0)), f (f (-5)), g (f (2)).

3. Find f/g and g/f and D, R, f (X) =2, g (x) =x2+1.

4. u(x)= 4x-5, v(x)=x2, f(x)=1/x find u(v(f(x))), v(u(f(x))), f(v(u(x))).


Limits of a functions:

F(x) approaches the limits L as x approach x so,









Note : f(x)= state whether f(x) is continuous :
F(2)=1


Since f(2) lim f(x)
F(x) is not continuous


Theory: if f and g be two continuous function at x=c then :
F+g, f-g, f/g are continuous function.

Limit laws : if L, M, C and K are real numbers and
Lim f(x)=L and Lim g(x)=M then :
1. sum rule lim (f(x)+g(x))=L+M
2. Difference rule lim (f(x)-g(x))=L-M
3. Product rule lim (f(x).g(x))=L.M
4. Constant multiple lim k.f(x)=k.L
5. Quotient rule lim f(x)/g(x)=L/M, M not equal zero.
6. power rule if r and s are integers then lim(f(x))r/s=Lr/s
The derivative of a function f with respect to x is f defined by :