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Question
If \[g\left( x \right) = x^2 + x - 2\text{ and} \frac{1}{2} gof\left( x \right) = 2 x^2 - 5x + 2\] is equal to
Options
\[2 x - 3\]
\[2 x + 3\]
\[2 x^2 + 3x + 1\]
2 \[x^2 - 3x - 1\]
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Solution
We will solve this problem by the trial-and-error method.
Let us check option (a) first. \[\text{If f}\left( x \right) = 2x - 3\]
\[\frac{1}{2}\left( gof \right)\left( x \right) = g\left( f\left( x \right) \right)\]
\[ = \frac{1}{2}g\left( 2x - 3 \right)\]
\[ = \frac{1}{2}\left[ \left( 2x - 3 \right)^2 + \left( 2x - 3 \right) - 2 \right]\]
\[ = \frac{1}{2}\left[ 4 x^2 + 9 - 12x + 2x - 3 - 2 \right]\]
\[ = \frac{1}{2}\left[ 4 x^2 - 10x + 4 \right]\]
\[ = 2 x^2 - 5x + 2\]
The given condition is satisfied by (a).
So, the answer is (a).
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