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 \(f(g(x))=x\)
\(=f\left( \frac{x+1}{2} \right)\)
\(=|2\left( \frac{x+1}{2} \right)-8|\)
\(=|x-7|\)
\(|x-7|=x\)
\(x=3.5\)
\(-(x-7)=x\)
\(7=2x\)
\(x=3.5\)