When finding the domain you ask the question:

Are there any numbers that cannot be substituted for x?

If the answer is "no", then the DOMAIN is 

If the answer is "yes", then exclude any such numbers from the DOMAIN.

When finding the range, replace f(x) by y.  Solve for x.

Ask the question:

Are there any numbers that cannot be substituted for y?

If the answer is "no", then the RANGE is 

If the answer is "yes", then exclude any such numbers from the RANGE.

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In your problem, ask the question:

Are there any numbers that cannot be substituted for x in f(x)=-2x-7?

The answer is "no" because there are no denominators to become 0,
and there are no square roots to have negative numbers under them.

So the domain is 

Replace f(x) by y

y = -2x-7
 Solve for x

2x = -y-7
 x = 

Ask the question:

Are there any numbers that cannot be substituted for y?

The answer is "no" because there are no denominators to become 0, since
the only denominator is the constant 2.  There are no square roots to have
negative numbers under them.

So the RANGE is 

Edwin