# What is the domain and range of #1/(x-7)#?

Domain: all real numbers x such that

Range: all real numbers.

The set of all values of x for which the function is defined is called the domain.

That includes all values of x for this function, excluding exactly 7, as that would result in a division by zero.

The set of all values y that the function is capable of producing is called the range.

It is the set of all real numbers in this instance.

It's time for a mental experiment:

The denominator of your function is 7 minus that number, or just the tiny number, if x is just a TINY bit greater than 7.

One can make y = f(x) as large as desired by selecting an input number x that is near to 7 but marginally greater than 7. This is because 1 divided by a tiny number equals a BIG number.

Choose an input number x that is close to 7, but just a little bit less than 7. This will give you y equal to 1 divided by a very tiny NEGATIVE number, resulting in a very large negative number. In fact, you can make y = f(x) be as big a NEGATIVE number as you want.

Another way to verify your reasoning is to graph the function: graph{1/(x-7) [-20, 20, -10, 10]}.

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The domain of ( \frac{1}{x-7} ) is all real numbers except for ( x = 7 ), because division by zero is undefined.

The range of ( \frac{1}{x-7} ) is all real numbers except ( y = 0 ), since the function approaches infinity as ( x ) approaches 7 from either direction.

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