# Determine whether `f(x)= sqrt(x+1) - 1` and `g(x) = (x+1)^2 -1` are inverse functions

embizze | High School Teacher | (Level 1) Educator Emeritus

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As justaguide has shown, the functions are inverses. However, you need to take into account the domain.

Since the domain of `f(x)=sqrt(x+1)-1` is x>-1, the domain for the inverse function g(x) will also be constrained. The domain for g(x) in this case is also x>-1.

The graph of f(x) is red, the graph of g(x) is blue:

justaguide | College Teacher | (Level 2) Distinguished Educator

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It has to be determined if `f(x)= sqrt(x+1) - 1` and `g(x) = (x+1)^2 -1` are inverse functions.

The functions f(x) and g(x) are inverse functions if `f(g(x)) = g(f(x)) = x`

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

=> `sqrt((x+1)^2 -1 + 1) - 1`

=> `sqrt((x+1)^2) - 1`

=> `x + 1 - 1`

=> x

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

=> `(sqrt(x+1) - 1+1)^2 -1`

=> `(sqrt(x+1))^2 -1`

=> `x + 1 - 1`

=> x

This proves that the functions `f(x)= sqrt(x+1) - 1` and `g(x) = (x+1)^2` are inverse functions.

lkballer24 | Student, Grade 11 | (Level 1) Valedictorian

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Sorry, the question is right but the real equations are..

1) f(x)=  √x+1 - 1

2) g(x)= (x+1) 2  -1

For number 1) By the way it is square root of x+1 then -1