SOLUTION: Given f(x)=x^3+2 and g(x)=cube_root(x-2), show that f(g(x))=x and g(f(x))=x. Graph both functions on the same set of coordinate axes.

Algebra ->  Graphs -> SOLUTION: Given f(x)=x^3+2 and g(x)=cube_root(x-2), show that f(g(x))=x and g(f(x))=x. Graph both functions on the same set of coordinate axes.       Log On


   

Question 113796: Given f(x)=x^3+2 and g(x)=cube_root(x-2), show that f(g(x))=x and g(f(x))=x. Graph both functions on the same set of coordinate axes.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#3

f%28x%29=x%5E3%2B2 Start with f(x)

f%28g%28x%29%29=%28g%28x%29%29%5E3%2B2 Plug in x=g%28x%29


f%28g%28x%29%29=%28root%283%2Cx-2%29%29%5E3%2B2 Replace each g%28x%29 with root%283%2Cx-2%29. This is given since g%28x%29=root%283%2Cx-2%29


f%28g%28x%29%29=%28x-2%29%2B2 Cube the cube root (these two functions "undo" each other and cancel out)


f%28g%28x%29%29=x Combine like terms


So this shows that f%28g%28x%29%29=x


g%28x%29=root%283%2Cx-2%29 Start with g(x)


g%28f%28x%29%29=root%283%2Cf%28x%29-2%29 Plug in x=f%28x%29


g%28f%28x%29%29=root%283%2Cx%5E3%2B2-2%29 Replace f(x) with x%5E3%2B2


g%28f%28x%29%29=root%283%2Cx%5E3%29 Combine like terms


g%28f%28x%29%29=x Take the cube root of x%5E3 to get x. Remember the cube root and the cube cancel each other out.




So this shows that g%28f%28x%29%29=x


Now if we graph the two functions, we get