Question

Find the domain of the function \(y=\frac{x+1}{x+2}\)

Short answer

The domain of the function \(y=\frac{x+1}{x+2}\) is all real numbers except for \(x=-2\).

Step-by-step solution

Define the Function

The function given in the problem is \(y=\frac{x+1}{x+2}\).

Identify the Restrictions

The only restriction to the domain of this function comes from the denominator of the rational function. Since the denominator cannot be equal to zero (as it leads to undefined or infinity in mathematics), this will set up our restriction.

Set the Denominator Equal to Zero

To find out the restriction, set the denominator equal to zero and solve for x: \[ x + 2 = 0 \] Solving this, we get \(x = -2\).

Determine the Domain

Since \(x = -2\) makes the expression undefined, it has to be excluded from the domain. Hence, the domain of the given function is all real numbers except -2. This can be written as \(-\infty < x < -2\) or \(-2 < x < \infty\).