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

Question

Isaiah Anthony · Accepted Answer

Domain: all real numbers x such that #x != 7#
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]}.

Jace Anderson · Answer

undefined 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.