Question

Find the range of \[f\left( x \right) = \operatorname{sgn} \left( {{x^2} - 2x + 3} \right)\].

Answer 1

Hint: First write the sign function in terms of number and its absolute function. After that find the nature of the quadratic equation and the value of a which will determine whether the absolute value will give a positive sign or negative sign. After that, the function will return the range.

Complete step-by-step answer:
Given:- \[f\left( x \right) = \operatorname{sgn} \left( {{x^2} - 2x + 3} \right)\].
A real number can be expressed as the product of its absolute value and its sign function
$y = \operatorname{sgn} \left( y \right) \times \left| y \right|$
The absolute value will also be expressed as the product of the real number and its sign function.
\[\left| y \right| = y \times \operatorname{sgn} \left( y \right)\]
Then, the sign function is the ratio of the absolute function to its real number.
$\operatorname{sgn} \left( y \right) = \dfrac{{\left| y \right|}}{y}$
Put $y = {x^2} - 2x + 3$ in the above equation,
$\operatorname{sgn} \left( {{x^2} - 2x + 3} \right) = \dfrac{{\left| {{x^2} - 2x + 3} \right|}}{{{x^2} - 2x + 3}}$
Now, find the nature of the quadratic equations by getting the value of discriminant of the equation,
$D = {b^2} - 4ac$
Here, put a=1, b=-2 and c=3. Then, the discriminant will be,
$D = {\left( { - 2} \right)^2} - 4 \times 1 \times 3$
Open the bracket to square the value and multiply the other numbers,
$D = 4 - 12$
Subtract 12 from 4 to get the value of discriminant,
$D = - 8$
Here, $a > 3$ and $D < 0$. It implies that the quadratic equation is always positive.
Thus, the absolute value will give a positive sign.
$\operatorname{sgn} \left( {{x^2} - 2x + 3} \right) = \dfrac{{{x^2} - 2x + 3}}{{{x^2} - 2x + 3}}$
Cancel out the common factors from the numerator and denominator,
$\operatorname{sgn} \left( {{x^2} - 2x + 3} \right) = 1$.
Thus,
$f\left( x \right) = 1$

Hence, the range of the function is 1.

Note: The students must remember the sign formula to solve this type of problem. He/she must also check the value of a and the nature of the quadratic equation to analyze whether the mode function will give a positive sign or negative sign. If he/she forgets any condition the range of the function will be wrong.