Determine the functions f∘g,g∘f,f∘f and g∘g and their domains if f(x)=x3+2 and g(x)=3√x
For f∘g
f∘g=f(g(x))Definition of f∘gf∘g=(3√x)3+2Definition of gf∘g=x+2Definition of f
The domain of the function is (−∞,∞)
For g∘f
g∘f=g(f(x))Definition of g∘fg∘f=3√(x3+2)Definition of g
We know that if the index is an odd number then the domain of function is (−∞,∞)
For f∘f
f∘f=f(f(x))Definition of f∘ff∘f=(x3+2)3+2Definition of ff∘f=x9+6x6+12x3+8+2Simplifyf∘f=x9+6x6+12x3+10Definition of f
The domain of the function is (−∞,∞)
For g∘g
g∘g=g(g(x))Definition of g∘gg∘g=3√3√xDefinition of gg∘g=6√xDefinition of g
We know that if the index is any even number, the radicand can't have a negative value. So the domain of function is [0,∞)
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