First slide

Algebra of real valued functions

Question

If $g (x) = x^{2} + x - 1$ and $(g o f) (x) = 4 x^{2} - 10 x + 5$ , then $f (\frac{5}{4})$ is equal to

Moderate

A

$- \frac{1}{2}$
B

$- \frac{3}{2}$
C

$\frac{3}{2}$
D

$\frac{1}{2}$

Solution

$W e h a v e g (f (x)) = f^{2} (x) + f (x) - 1$
$\Rightarrow g (f (\frac{5}{4})) = f^{2} (\frac{5}{4}) + f (\frac{5}{4}) - 1 \to (1)$
$A l s o$
$g (f (\frac{5}{4})) = 4 {(\frac{5}{4})}^{2} - 10. \frac{5}{4} + 5 = - \frac{5}{4} \to (2)$
$B y (1) a n d (2) w e h a v e$
$- \frac{5}{4} = f^{2} (\frac{5}{4}) + f (\frac{5}{4}) - 1$
$\Rightarrow {(f (\frac{5}{4}) + \frac{1}{2})}^{2} = 0$
$\Rightarrow f (\frac{5}{4}) = - \frac{1}{2}$

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