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).