How do you find (f of g of h) if f(x)=x^2+1f(x)=x2+1 g(x)=2xg(x)=2x and h(x)=x-1h(x)=x−1?

1 Answer
George C.
Mar 4, 2018

(f@g@h)(x) = 4x^2-8x+5(f∘g∘h)(x)=4x2−8x+5

Explanation:

Given:

{ (f(x) = x^2+1), (g(x) = 2x), (h(x) = x-1) :}

One way of thinking about these function compositions is to go back and forth between the symbols and verbal descriptions of what the functions do.

In our example:

  • f takes the square of a number and adds 1

  • g doubles a number

  • h subtracts 1 from a number

So a verbal description of the composed f@g@h as a sequence of steps might be:

  • Subtract 1

  • Double

  • Square

  • Add 1

So in symbols we might describe this process thus:

x -> x-1 -> 2(x-1) -> (2(x-1))^2 -> (2(x-1))^2+1

So:

(f@g@h)(x) = f(g(h(x)))

color(white)((f@g@h)(x)) = (2(x-1))^2+1

color(white)((f@g@h)(x)) = 4(x^2-2x+1)+1

color(white)((f@g@h)(x)) = 4x^2-8x+4+1

color(white)((f@g@h)(x)) = 4x^2-8x+5

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