Questions: Find all numbers for which the rational expression is undefined. If the rational expression is defined for all real numbers, state this. (y+3)/(y^2-9) Type the values for which the rational expression is undefined. Select the correct choice below and fill in any answer boxes within your A. (Use a comma to separate answers as needed.) B. The rational expression is defined for all real numbers.

Solution Steps

To determine when the rational expression \(\frac{y+3}{y^2-9}\) is undefined, we need to find the values of \(y\) that make the denominator zero. The denominator is \(y^2 - 9\), which is a difference of squares and can be factored as \((y-3)(y+3)\). The expression is undefined when the denominator equals zero, so we solve the equation \(y^2 - 9 = 0\) to find these values.

We are given the rational expression

\[ \frac{y+3}{y^2-9} \]

and need to determine the values of \(y\) for which this expression is undefined.

The expression is undefined when the denominator equals zero. Therefore, we solve the equation:

\[ y^2 - 9 = 0 \]

The denominator can be factored as:

\[ y^2 - 9 = (y-3)(y+3) \]

Setting each factor equal to zero gives us:

\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \] \[ y + 3 = 0 \quad \Rightarrow \quad y = -3 \]

Thus, the values of \(y\) that make the rational expression undefined are \(y = -3\) and \(y = 3\).

Final Answer