1. Functions & Graphs by Mrs. Sujata Tapare Prof. Ramkrishna More A.C.S. College, Akurdi

3. Function terminology R Z f 4.3 4 Domain Co-domain Pre-image of 4 Image of 4.3 f maps R to Z f(4.3)

4. Even more functions Not a function Range 1 2 3 4 5 “ a” “ bb“ “ cccc” “ dd” “ e” 1 2 3 4 5 a e i o u It is a function

6. 5 5 0 -2 Constant function: f(x) = 5 Rational function: f(x) = (4- x 2 )/(x+2) Polynomial function: f(x) = x 2 Identity function: f(x) = x

9. One-one functions Function f : R -> R is one-one for any x, y Є R, f(x) = f(y)  x = y. Examples Let f:R  R defined by, f(x) = 3x – 2 f(x)= f(y)  3x – 2 = 3y – 2  x = y Thus, f(x) is one-one Let f:R  R defined by, f(x) = 3x 2 – 2 f(x)= f(y)  3x 2 –2= 3y 2 –2  3x 2 = 3y 2  x = ± y Thus, f(x) is not one-one

14. Compositions of functions g f f ○ g g(a) f(a) (f ○ g)(a) g(a) f(g(a)) a A B C

17. Floor:  x  -5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5

21. o

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27. f(a) = f(b) f’(c 1 )=0 f’(c 2 ) = 0 c 1 c 2 a 0 b

28. o c 1 b a c 2