The number of value/values of x for which sin `y=x^(2)-2x` si possible is

Options

A. 0
B. 1
C. infinite
D. none of these

Explanation

To solve the problem of finding the number of values of \( x \) for which \( \sin y = x^2 - 2x \) is possible, we will follow these steps:

### Step 1: Understand the Equation
We start with the equation:
\[ \sin y = x^2 - 2x \]
We need to determine for which values of \( x \) this equation holds true.

### Step 2: Rewrite the Quadratic Expression
We can rewrite the quadratic expression \( x^2 - 2x \) as:
\[ x^2 - 2x = (x - 1)^2 - 1 \]
This transformation helps us analyze the range of the expression more easily.

### Step 3: Determine the Range of the Quadratic Expression
From the rewritten form, we have:
\[ \sin y = (x - 1)^2 - 1 \]
The term \( (x - 1)^2 \) is always non-negative (i.e., \( \geq 0 \)). Therefore, the minimum value of \( (x - 1)^2 \) is 0, which occurs when \( x = 1 \).

Thus, the minimum value of \( (x - 1)^2 - 1 \) is:
\[ 0 - 1 = -1 \]
As \( (x - 1)^2 \) can take any non-negative value, \( (x - 1)^2 - 1 \) can take any value greater than or equal to \(-1\).

### Step 4: Compare with the Range of the Sine Function
The sine function, \( \sin y \), has a range of:
\[ [-1, 1] \]
This means \( \sin y \) can take values from \(-1\) to \(1\).

### Step 5: Determine Valid Values of \( x \)
For \( \sin y = (x - 1)^2 - 1 \) to hold, we need:
\[ -1 \leq (x - 1)^2 - 1 \leq 1 \]
This simplifies to:
1. \( (x - 1)^2 - 1 \geq -1 \) which is always true since it simplifies to \( (x - 1)^2 \geq 0 \).
2. \( (x - 1)^2 - 1 \leq 1 \) simplifies to:
\[ (x - 1)^2 \leq 2 \]
Taking the square root, we find:
\[ -\sqrt{2} \leq x - 1 \leq \sqrt{2} \]
Therefore:
\[ 1 - \sqrt{2} \leq x \leq 1 + \sqrt{2} \]

### Step 6: Conclusion
The interval \( [1 - \sqrt{2}, 1 + \sqrt{2}] \) contains infinitely many values of \( x \). Therefore, the number of values of \( x \) for which \( \sin y = x^2 - 2x \) is possible is infinite.

### Final Answer
The number of values of \( x \) for which \( \sin y = x^2 - 2x \) is possible is **infinite**.

---