What is the domain and range of #(x^4 + x^2) /(1+x^2)#?

Considering

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) that the function can take.

For the function ( \frac{x^4 + x^2}{1 + x^2} ), the domain consists of all real numbers except those that make the denominator equal to zero. Since the denominator ( 1 + x^2 ) can never be zero for any real value of x, the domain of the function is all real numbers, or ( (-\infty, +\infty) ).

To find the range of the function, we need to analyze its behavior as ( x ) approaches positive or negative infinity. As ( x ) approaches positive or negative infinity, the term ( x^2 ) dominates the expression, and the function behaves like ( \frac{x^4}{x^2} ), which simplifies to ( x^2 ). Since ( x^2 ) can take all non-negative real values, the range of the function is ( [0, +\infty) ).