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Composite Functions

Composite functions
  • If you have a function, \(f(g(x))\), then you do the inside most function first and then work outwards.
  • If we had \(g(x)=2x+1\) and \(f(x)=x^2+1\), then \(f(g(x))\) would be the same as \(f(2x+1)\) or \((2x+1)^2+1\).
Modulus with composite functions
  • \(f(x)=|2x-8|\)
  • \(g(x)=\frac{x+1}{2}\)
  • Find \(fg(3)\):
    \(=f\left( \frac{3+1}{2} \right)\)
    \(=f(2)\)
    \(=|2\times2-8|\)
    \(=|-4|\)
    \(=4\)
  • Find \(f(g(x))=x\)
    \(=f\left( \frac{x+1}{2} \right)\)
    \(=|2\left( \frac{x+1}{2} \right)-8|\)
    \(=|x-7|\)
    \(|x-7|=x\)
    \(x=3.5\)
    \(-(x-7)=x\)
    \(7=2x\)
    \(x=3.5\)
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