Prove that two equations are inverses of each other.

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To prove that the functions f(x) = 9 - x^2 and g(x) = sqrt(9 - x) are inverses, it is necessary to show that f(g(x)) = x and g(f(x)) = x. The calculation for f(g(x)) simplifies to 9 - (9 - x), which equals x, confirming one direction of the inverse relationship. Additionally, it is important to compute g(f(x)) to complete the proof. The discussion emphasizes the correct terminology, noting that functions, not equations, are referred to as inverses. This verification process is crucial in understanding the relationship between the two functions.
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Homework Statement


f(x) = 9-x^2, x ≥ 0. g(x) = sqrt (9-x)

Homework Equations


Two equations are inverses of each other if f(g(x)) = g(f(x)) = x.

The Attempt at a Solution



f(g(x)) = 9 - (sqrt (9-x))^2

= 9 - (9-x)
 
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Open the brackets?
 
Hmm?
 
Physics-Pure said:

Homework Statement


f(x) = 9-x^2, x ≥ 0. g(x) = sqrt (9-x)

Homework Equations


Two equations are inverses of each other if f(g(x)) = g(f(x)) = x.

The Attempt at a Solution



f(g(x)) = 9 - (sqrt (9-x))^2

= 9 - (9-x)
So close! Simplify the above.

Don't forget that you also have to show that g(f(x)) = x.
 
BTW, we don't normally say that equations are inverses of each other - we say that functions are inverses of each other.
 

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