Verify that the functions f and g, are inverses of each other by showing that f(g(x))= x and g(f(x))= x. Graph both functions on the same graph.f(x)= (x +1)^2; x>= -1 g(x)= sqrt(x) -1

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Given `f(x)=(x+1)^2` and `g(x)=sqrt(x)-1` ; show that `f(x) "and" g(x)` are inverses of each other.

(1) We can show that two functions are inverses by showing that `f(g(x))=g(f(x))=x`

(2) `f(g(x))=(sqrt(x)-1+1)^2=(sqrt(x))^2=x`

(3) `g(f(x))=sqrt((x+1)^2)-1=x+1-1=x`

Thus the functions are inverses.

The graphs:

Note that the graph of `f(x)` starts at -1; for the functions to be inverses the domain of `f` is the range of `g` , thus the domain of `f` is `x>=-1` . The domain of `g` is the range of `f` , so the domain of `g` is `x>=0` . (The additional restriction on `g` , that is that the argument of the square root be nonnegative is covered)

Note also that the graphs are symmetric about the line y=x.