Question

What is the domain and range of:$y = \sin \left( {{x^2} - 3x + 2} \right)$
Domain =R: Range=$\left[ { - \dfrac{1}{2},\dfrac{1}{2}} \right]$
Domain=R: Range=$\left[ { - 1,1} \right]$
Domain=R: Range=$\left[ {0,1} \right]$
None of these

Answer 1

Hint : In this question a function is given so we will substitute the value of x in the function and we will check for the domain and range of the function.

Complete step-by-step answer :
$y = \sin \left( {{x^2} - 3x + 2} \right)$
Let the function be
$f\left( x \right) = \sin \left( {{x^2} - 3x + 2} \right)$
Now we substitute the value any of x in the function to find the domain of the function
When x=2
Hence

$$f\left( 2 \right) = \sin \left( {{2^2} - 3 \times 2 + 2} \right) \\\ = \sin (0) \\\ = 0 \;$$

When x=-4

$$f\left( { - 4} \right) = \sin \left( {{{\left( { - 4} \right)}^2} - 3 \times \left( { - 4} \right) + 2} \right) \\\ = \sin (16 + 12 + 2) \\\ = \sin \left( {30} \right) \\\ = 0.5 \;$$

When $x = \pi$

$$f\left( \pi \right) = \sin \left( {{\pi ^2} - 3 \times \left( \pi \right) + 2} \right) \\\ = \sin (2.44) \\\ = 0.042 \;$$

Now from the above calculation we can say whenever a real value of x is substituted in the function $f\left( x \right) = \sin \left( {{x^2} - 3x + 2} \right)$, it gives a real number
Hence we can say the domain of the function $y = \sin \left( {{x^2} - 3x + 2} \right)$ is real number R.
Now in the function $y = \sin \left( {{x^2} - 3x + 2} \right)$, the function is a sine function and the range of the sine function is $\left[ { - 1,1} \right]$.
$- 1 \leqslant \sin \theta \leqslant 1$
Now in the function $y = \sin \left( {{x^2} - 3x + 2} \right)$as already observed above if we substitute any real number in the function it gives a real number which lies in the range $\left[ { - 1,1} \right]$.
Hence we can say the range of the function $y = \sin \left( {{x^2} - 3x + 2} \right)$is $\left[ { - 1,1} \right]$.
Therefore the domain of the function $y = \sin \left( {{x^2} - 3x + 2} \right)$ is R and the range is$\left[ { - 1,1} \right]$.
So, the correct answer is “Option C”.

Note : Students must note that the domain is the set of all possible values of x for which the function f(x) will be defined and the range refers to the possible range of values that the function f(x) can attain for those values of x which are in the domain of f(x).