How Do You Find (f Of G Of H) If F(x)=x^2+1 G(x)=2x And H(x)=x-1?

How do you find (f of g of h) if #f(x)=x^2+1# #g(x)=2x# and #h(x)=x-1#? Precalculus Functions Defined and Notation Function Composition

1 Answer

George C. Mar 4, 2018

#(f@g@h)(x) = 4x^2-8x+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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