Question

Determine the domain and range Express your answer in interval notation

Answer 1

The domain of a function is the set of all values that the x-variable can take.

On the other hand, the range of the function is the set of all values that the function takes when it is evaluated at elements of the domain.

For the given expression:

$p(x)=-\frac{1}{(x-1)^2}$

The denominator is (x-1)^2. Since the denominator must be different from 0, then:

$\begin{gathered} (x-1)^2\ne0 \\ \Rightarrow x-1\ne0 \\ \Rightarrow x\ne1 \end{gathered}$

Then, the only restriction for the variable x is not to be equal to 1. Then, the domain of p(x) is the set of all real numbers except 1, which can be written using interval notation as:

$(-\infty,1)\cup(1,\infty)$

Since the exponent of the denominator is 2, then the denominator is always positive. Since the coefficient of the term 1/(x-1)^2 is -1, then the whole expression must always be negative. Additionally, there is no way in which the expression can be equal to 0.

Then, the range of the function is the set of all negative numbers, which can be expressed using interval notation as:

$(-\infty,0)$

Therefore, the answers are:

$\begin{gathered} \text{ Domain: }(-\infty,1)\cup(1,\infty) \\ \\ \text{ Range: }(-\infty,0) \end{gathered}$