Question

What is the domain of y equals StartFraction x squared plus 2 over x cubed minus x EndFraction?

Answer options with 5 options
A.
all real numbers except StartRoot 2 EndRoot

B.
all real numbers except negative StartRoot 2 EndRoot

C.
all real numbers except 0

D.
all real numbers except negative 1 and 1

E.
all real numbers except negative 1, 0, and 1

Answer 1

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:

1. \( x = 0 \)
2. \( x - 1 = 0 \) → \( x = 1 \)
3. \( x + 1 = 0 \) → \( x = -1 \)

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.**