Solving for ( f(x) ) Given ( frac{x-1}{x 1} x^2 )
When presented with the function ( frac{x-1}{x 1} x^2 ) , the objective is often to express ( f(x) ) in terms of ( x ). This task involves a series of algebraic manipulations that lead to a general solution. Let's explore the steps in detail.
Step-by-Step Solution
1. Finding ( f(5) )
Let's begin by finding ( f(5) ). The initial step is to equate ( frac{x-1}{x 1} 5 ) . Solve for ( x ):
( frac{x-1}{x 1} 5 ) ( x-1 5(x 1) ) ) ( x-1 5x 5 ) ) ( -4 4x ) ) ( x -1.5 ) )
Now, substitute ( x -1.5 ) into the original equation ( f(t) t^2 ):
( f(5) (-1.5)^2 2.25 ) )
2. Generalizing for ( f(n) )
To find ( f(n) ) for some number ( n ), follow the same procedure:
( frac{x-1}{x 1} n ) ( x-1 n(x 1) ) ) ( x-1 nx n ) ) ( x -n-1 ) ) ( x -frac{n 1}{n-1} ) )
Now, substitute ( x -frac{n 1}{n-1} ) back into the original equation ( f(t) t^2 ) to find ( f(n) ):
( f(n) left( -frac{n 1}{n-1} right)^2 left( frac{n 1}{n-1} right)^2 ) ) ( f(n) frac{(n 1)^2}{(n-1)^2} )
3. Simplified Expression for ( f(x) )
The simplified expression for ( f(x) ) is:
( f(x) left( frac{x 1}{x-1} right)^2 )
Alternative Approaches
Here are a few additional insights into the problem through various methods:
Method 1: Direct Substitution
If you know that ( frac{x-1}{x 1} y ), then:
( x frac{y 1}{y-1} )
Therefore, ( f(y) y^2 ) and replacing ( y ) with ( x ) gives:
( f(x) left( frac{x 1}{x-1} right)^2 ) )
Method 2: Componendo and Dividendo
Using the method of componendo and dividendo:
( frac{x-1}{x 1} t ) ( frac{x-1 x 1}{x-1-(x 1)} frac{t 1}{t-1} ) ( x frac{t 1}{t-1} ) ( f(t) t^2 ) ( f(x) left( frac{x 1}{x-1} right)^2 )
Method 3: Solving for ( x )
Let ( u frac{x-1}{x 1} ), then solve for ( x ):
( x frac{u 1}{u-1} )
Substitute ( x ) back into the function ( f(u) u^2 ) and replace ( u ) with ( x ):
( f(x) left( frac{x 1}{x-1} right)^2 ) )
Conclusion
The function ( f(x) ) given by ( frac{x-1}{x 1} x^2 ) can be expressed as:
( f(x) left( frac{x 1}{x-1} right)^2 ) )
This result was derived through algebraic manipulation and substitution. It is a prime example of how to solve such problems step-by-step and ensure that all steps are faithful to the original equation.