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6.1 continued… Composition of Functions

Learn how to compose functions to create a chain of operations for transforming inputs into outputs. Examples and step-by-step instructions included.

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6.1 continued… Composition of Functions

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  1. 6.1 continued…Composition of Functions

  2. Remember what a function does input output FIRST LETTER Dog breed output

  3. FIRST LETTER G D C

  4. ADD 2 5 7 -4 -2 +2

  5. SQUARE IT 5 25 -4 16

  6. x y x f(x) f( )

  7. BUT WAIT!!! There’s more!!! You can even make a whole chain of functions and turn it into one big {composite} function machine… MULTIPLY BY 5 ADD 2 1 3 15 4 6 30 -10 -8 -40

  8. How do we build a function machine? Remember… f(x) = 3x + 2 f(g(x)) = 3(g(x)) + 2 f(7) = 3(7) + 2 Note: this is sometimes written as 3(11) + 2 f(11) = f( ) = 3( ) + 2 This DOES NOT mean multiply!!! It means find f(g(x)). It is read as “f of g of x” f( ) = 3( ) + 2 f( ) = 3( ) + 2

  9. Example 1: If f(x) = 2x and g(x) = x2 – 3x + 2 and h(x) = –3x – 4 then find each value. a) f[g(–3)] b) g[h(2)] c) h[f(–2)]

  10. Example 2 B. Find [f ○ g](x) and [g ○ f](x) for f(x) = 3x + 4 and g(x) = 2x – 1.

  11. Example 3 B. Find [f ○ g](x) and [g ○ f](x) for f(x) = 3x2 – x + 4 and g(x) = 2x – 1.

  12. 1) (h ◦ g)(x) 2) f(h(x)) 4) (f ◦ g)(x) 3) (j◦ h)(x)

  13. 2) f(h(x)) 1) (h ◦ g)(x) h(g(x))= 2[x – 1]+ 5 h()= 2[ ]+ 5 f(h(x))= [2x + 5]2 f()= [ ]2 h(g(x))= 2x – 2 + 5 f(h(x))= (2x + 5)(2x + 5) h(g(x))= 2x + 3 f(h(x))= 4x2 + 20x + 25 4) (f ◦ g)(x) 3) (j◦ h)(x) j(h(x))= 6[2x + 5] f(g(x))= [x-1]2 j()= 6[ ] j(h(x))= 12x + 30 f(g(x))= x2 – 2x +1

  14. The super-duper awesome amazing problem: g(f(h(j(x)))) [[2[6x]+5]2] – 1 [12x+5]2] – 1 [(12x+5)(12x+5)] – 1 [144x2 + 120x + 25] – 1 144x2 + 120x + 24

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