Find the domain of the function $$f(x) = \frac{1}{(x-1)^2+(x+1)^3}.$$ Express your answer in interval notation.

Answer 1

The function $f(x)$ is defined as long as the denominator is not equal to zero. To find the values of $x$ that make the denominator zero, we set $(x-1)^2+(x+1)^3=0.$ Since each term of $(x+1)^3$ is nonnegative for all real $x$, to make $(x-1)^2+(x+1)^3$ zero, we must have $(x-1)^2=0.$ This condition is satisfied if and only if $x-1=0,$ so $x=1.$

Thus, the domain of the function $f(x)$ is the set of all real numbers $x$ except $x=1,$ which we write in interval notation as $\boxed{(-\infty, 1) \cup (1, \infty)}.$