### Instruction: Find the range of the function
\[h(x) = \frac{2(x + 7)(x - 3)}{x + 7}.\].### Response: I want to find the set of possible outputs of this function, so I need to look at how it behaves for different values of x.
The first thing I notice is that the function is undefined when x = -7, since that would make the denominator zero.
So I can't include x = -7 in the domain of the function.
The next thing I notice is that the function simplifies to h(x) = 2(x - 3) when x is not equal to -7, since the factor of x + 7 cancels out in the numerator and denominator.
So the function is really just a linear function with a slope of 2 and a y-intercept of -6, except for a hole at x = -7.
A linear function has a constant rate of change, so it can take on any value as x increases or decreases, except for the hole.
So the range of the function is all real numbers, except for one value that corresponds to the hole.
To find that value, I can plug in x = -7 into the simplified function, h(x) = 2(x - 3).
I get h(-7) = 2(-7 - 3) = -20.
So the function never outputs -20, because that's where the hole is.
Therefore, the range of the function is all real numbers except -20.
I can write this using set notation as
\[\{y \in \mathbb{R} \mid y \neq -20\}.\]