To find the domain of the function \( y = \frac{x^2 + 2}{x^3 - x} \), we need to determine where the function is undefined. A rational function is undefined when its denominator is equal to zero.
The denominator of the function is \( x^3 - x \). We can factor this expression:
\[ x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1) \]
Now we set the denominator equal to zero to find the values of \( x \) that make the expression undefined:
\[ x(x - 1)(x + 1) = 0 \]
This gives us the values:
- \( x = 0 \)
- \( x - 1 = 0 \) → \( x = 1 \)
- \( x + 1 = 0 \) → \( x = -1 \)
Thus, the function is undefined at \( x = -1, 0, \) and \( 1 \).
Therefore, the domain of the function is all real numbers except \( -1, 0, \) and \( 1 \).
The correct answer is:
E. all real numbers except negative 1, 0, and 1.
1 answer