Question
A function $f$ is defined by $f(x)=3-2 x$. Find $x$ such that $f\left(x^2\right)=(f(x))^2$
✓
Answer
$f(x) = 3 – 2x f(x^2) = 3 – 2 (x^2) = 3 – 2x^2$
$(f(x))^2 = (3 – 2x)^2 = 9 + 4x^2 – 12x$
But
$f(x^2) = (f(x))^2 3 – 2x^2 = 9 + 4x^2 – 12x –2x^2 – 4x^2+ 12x + 3 – 9$
$= 0 –6x^2 + 12x – 6 = 0 (÷ by – 6)$
$\Rightarrow x^2 – 2x + 1 = 0 (x – 1) (x – 1) = 0 x – 1 = 0$ or $x – 1 = 0 x = 1$ The value of $x = 1$
$(f(x))^2 = (3 – 2x)^2 = 9 + 4x^2 – 12x$
But
$f(x^2) = (f(x))^2 3 – 2x^2 = 9 + 4x^2 – 12x –2x^2 – 4x^2+ 12x + 3 – 9$
$= 0 –6x^2 + 12x – 6 = 0 (÷ by – 6)$
$\Rightarrow x^2 – 2x + 1 = 0 (x – 1) (x – 1) = 0 x – 1 = 0$ or $x – 1 = 0 x = 1$ The value of $x = 1$
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