Function Composition

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1. f(x)=x^2. g(x)=x+1
   1. Find (f\circ g)(x)
      Solution
      (x+1)^2 or x^2+2x+1
   2. Evaluate (f\circ g)(-3)
      Solution
      4
   3. Find (g\circ f)(x)
      Solution
      x^2+1
   4. Find f\left(g\left(f(x)\right)\right)
      Solution
      (x^2+1)^2 or x^4+2x^2+1
2. f(x)=x^2. g(x)=\sqrt{x}
   1. Domain of h(x)=\left(f\circ g\right)(x)?
      Solution
      x\geq0
   2. Range of h(x)?
      Solution
      y\geq0
   3. Evaluate h(-4)
      Solution
      Undefined
3. h(x)=\sqrt{x+2}. Write h(x) as a composition of two functions f(x) and g(x)
   Solution
   f(x)=x+2. g(x)=\sqrt{x}. h(x)=g\left(f(x)\right)
4. h(x)=e^{2x-3}+\frac{1}{2x-3}. Write h(x) as a composition of two functions
   Solution
   f(x)=2x-3. g(x)=e^x+\frac{1}{x}. h(x)=g\left(f(x)\right)
5. Challenge: h(x)=2\left(e^{\log\left(\frac{x}{2}\right)}\right)^2+\left(e^{\log \left(\frac{x}{2}\right)} \right)-3
   Given h(x)=\left(a \circ b \circ c \circ d \right)(x), find the functions a, b, c, d
   Solution
   a(x)=2x^2+x-3. b(x)=e^x. c(x)=\log x. d(x)=\frac{x}{2}

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