# The above is just one example of how we can build new

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using a similar algebra. The above is just one example of how we can build new functions (total blanket cost) from other functions (separate discounts). The Algebra of Functions If f and g are functions, then ± ² ( ) ( ) ( ) f g x f x g x r r ± ² ( ) ( ) ( ) fg x f x g x ( ) ( ) ( ) f f x x g g x § · ¨ ¸ © ¹ ± ² ± ² ( ) ( ) f g x f g x 10 to F purchase price DCP p o.lp Tlp p lo p 1 0.1 0.9 p C p DCTCp czCpl TCDCp Glp o 9 p lo GCP o 9p 10 0.9 p 9 Calico 9 too ios 9400 0.94100 g 90 10 80 90 9 81

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Example 2 Let 1 ( ) 3 f x x ³ and ( ) g x x . Find the domains f and g and then find. (a) f g ´ (b) f g ³ (c) fg (d) f g (e) f g (f) f g (g) g f (h) f f (i) g g Example 3 Find f g h if ( ) 1 x f x x ´ , 20 ( ) g x x and ( ) 5 h x x ´ Going forward is one thing. Going backwards is yet another. Example 4 Decompose the function 3 ( ) 8 h x x ³ into two non-trivial functions f and g such that h f g . How many unique decompositions can you find? Df x x 433 Dg xlxzo x2t3x hCx7 Hx7gC G fCxltgCx hCxl fCxl glx has hlx7 ztTx uld Tx next Tx gox ii i i sameas a on xlxzo 93 9 g text _hCx hCx fI X 3 y Dh xlx 33 T f glum SPUD Decompose into 2 functions glx f x hCx T z T f gCxD hCx h x 1 x3t tD2 h x x 3 2 26 3 2 1
If you’ve ever done something you wish you could undo, then you are alone. I’m kidding, of course. We all have done something we’d like to have back. Although we can’t do that in our own lives, in the world of functions, this is

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• Fall '19
• Inverse function, Injective function

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