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Physics-Pure
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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)
So close! Simplify the above.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)
Two equations are considered inverses of each other if they can be manipulated to cancel out each other's effects and ultimately result in the original variable. In other words, when the equations are solved together, they "undo" each other.
To prove that two equations are inverses of each other, you can use the substitution method. This involves solving one of the equations for a variable and then substituting that expression into the other equation. If the resulting equation simplifies to a true statement, then the two equations are inverses of each other.
Yes, two equations can be inverses of each other even if they have different variables. The key is that the equations must be able to be manipulated to cancel out each other's effects and result in the original variable.
No, the order in which the equations are solved does not matter as long as they are both used in the same way. This means that if one equation is solved for the variable and substituted into the other equation, the same process should be applied if the equations are switched.
Proving that two equations are inverses of each other can be useful in solving systems of equations, finding the inverse of a function, and understanding the relationships between different mathematical operations. It also helps to check the accuracy of answers and identify potential errors in mathematical calculations.