Domain Analysis of (8+8x)/(x+6)²: Finding Valid Input Values

Given the following function:

$\frac{8+8x}{(x+6)^2}$

Does the function have a domain? If so, what is it?

To determine the domain of the function $\frac{8+8x}{(x+6)^2}$, we must find the values of $x$ that make the function undefined.

This function is a rational function with the numerator $8+8x$ and the denominator $(x+6)^2$. A rational function is undefined where its denominator is equal to zero.

Therefore, we need to solve for $x$ where the denominator equals zero:

• Set the denominator equal to zero: $(x+6)^2 = 0$.
• Take the square root of both sides, resulting in $x+6 = 0$.
• Solve for $x$ by subtracting 6 from both sides: $x = -6$.

This calculation shows that the function is undefined when $x = -6$. Therefore, the domain of the function includes all real numbers except $x = -6$.

Thus, the domain of the function is all real numbers except $x \ne -6$.

The correct choice is:

Yes, $x\ne-6$