Given ( f(x) = \frac{x-1}{x-2} ) and ( g(x) = x^2 ). Solve for ( f(g(x)) = 5 )
Answer:
To solve for ( f(g(x)) = 5 ), we first need to find an expression for ( f(g(x)) ):
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Calculate ( g(x) ):
( g(x) = x^2 ) -
Substitute ( g(x) ) into ( f(x) ):
$$ f(g(x)) = f(x^2) = \frac{x^2 - 1}{x^2 - 2} $$ -
Set up the equation ( f(g(x)) = 5 ):
$$ \frac{x^2 - 1}{x^2 - 2} = 5 $$ -
Solve for ( x ):
Multiply both sides by ( x^2 - 2 ):
$$ x^2 - 1 = 5(x^2 - 2) $$
$$ x^2 - 1 = 5x^2 - 10 $$
Bring all terms to one side of the equation to set it to zero:
$$ x^2 - 1 - 5x^2 + 10 = 0 $$
$$ -4x^2 + 9 = 0 $$
$$ 4x^2 = 9 $$
$$ x^2 = \frac{9}{4} $$
$$ x = \pm \frac{3}{2} $$ -
Check the solutions:
- ( x = \frac{3}{2} )
- ( x = -\frac{3}{2} )
The solution set is: ( \left{ -\frac{3}{2}, \frac{3}{2} \right} ), which corresponds to option (I).
Therefore, the correct answer is (I) ( \left{ -\frac{3}{2}, \frac{3}{2} \right} ).