Explain exponential functions

Explain exponential functions Exponential funcuon v B"'tive constant nutnber. called '101' 'ion. argument adopts anv values. the tunenon values are otherwise We will have a multi-valncel iunction. So. the tonction y • has at 1/4 tour dinerc•nt values y — .Ai. But we consider as the function value only y A. Graphs of an exm»ncnnal timetion tot 2 and a 112 are shown on Fig. l. 18. All they are going through the point (0. I At a = I we have as a graph a straight line, parallel to A-axis. i.e. the function becomes a constant value. equal to l. At a an exi»nential tuncuon increases. and at O < a I —decreases. x

E x p o n e n t i a l f u n c t i o n i s d e f i n e d a s f : ℝ \to ℝ^{+} f (x) = a^{x}, w h e r e a > 0 a n d a \neq 1 .' a' i s a c o n s \tan t a n d a \in ℝ E . g y = 2^{x}, y = 10^{x}, e t c T h e t e x t a c t u a l l y s a y s t h a t C o n s i d e r a n e x a m p l e f (x) = 81^{x} f (\frac{1}{4}) = 81^{\frac{1}{4}} N o w 81^{\frac{1}{4}} c a n t a k e m a n y v a l u e s l i k e 3, 3 i, - 3 i, - 3 B u t \sin c e c o - d o m a i n o f f u n c t i o n i s p o s i t i v e r e a l n u m b e r s s o, w e w i l l o n l y c o n s i d e r p o s i t i v e r e a l v a l u e s i . e . 3 h e r e f (\frac{1}{4}) = 81^{\frac{1}{4}} = 3 I f a = 1, t h e n n o m a t t e r w h a t i s t h e v a l u e o f x, f (x) = 1 H o w e v e r n o t e i n e x p o n e n t i a l f u n c t i o n a > 0 a n d a \neq 1

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