If f(x) + f((x - 4)/(x - 3)) = x^2, ∀ x ∈ ℝ - {3}, then f(2) is equal to?
Understand the Problem
The question is asking to find the value of the function f(2) based on the given expression involving f(x) and its properties. It requires understanding functions and possibly algebraic manipulation.
Answer
The value of \( f(2) \) is \( 2 \).
Answer for screen readers
The value of ( f(2) ) is ( 2 ).
Steps to Solve
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Understand the equation given The equation provided is $$ f(x) + f\left(\frac{x-4}{x-3}\right) = x^2 $$ for all $x$ in the set of real numbers excluding 3.
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Substitute x=2 to find f(2) We replace $x$ with 2 in the equation: $$ f(2) + f\left(\frac{2-4}{2-3}\right) = 2^2 $$ Calculate the argument of the second function: $$ \frac{2-4}{2-3} = \frac{-2}{-1} = 2 $$
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Rewrite the equation Substituting this value back into the equation gives: $$ f(2) + f(2) = 4 $$
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Combine like terms This simplifies to: $$ 2f(2) = 4 $$
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Solve for f(2) To find $f(2)$, divide both sides by 2: $$ f(2) = 2 $$
The value of ( f(2) ) is ( 2 ).
More Information
This problem involves function properties and algebraic manipulation, specifically how to use substitution and simplification to solve for an unknown function value. The result shows that ( f(2) ) is a specific integer.
Tips
- Forgetting to properly substitute into both parts of the function equation.
- Miscalculating the value of ( \frac{x-4}{x-3} ) when substituting different values of ( x ).
- Overlooking the need to simplify the equation correctly before solving for ( f(2) ).
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