If the range of \( f(x) \) is given by \( (-\infty, 1] \), what is the range of \( |f(x)| \) ? The range of \( |f(x)| \) is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

To find the range of \( |f(x)| \), we need to consider the effect of the absolute value function on the range of \( f(x) \). Given that the range of \( f(x) \) is \( (-\infty, 1] \), this means that \( f(x) \) can take on any value less than or equal to 1, but it can also take on negative values because the range includes \( -\infty \). The absolute value function, \( |x| \), always returns a non-negative value. It does this by taking the positive value of \( x \) if \( x \) is non-negative, and the negative value of \( x \) if \( x \) is negative. So, for \( |f(x)| \), we take the absolute value of every possible value of \( f(x) \) from its range \( (-\infty, 1] \): - If \( f(x) \) is negative or zero, \( |f(x)| \) will be positive or zero, respectively. - If \( f(x) \) is positive, \( |f(x)| \) will be the same as \( f(x) \). Since \( f(x) \) can be as large as 1, \( |f(x)| \) can also be as large as 1. However, since \( f(x) \) can be any negative number or zero, \( |f(x)| \) will be at least 0. Therefore, the range of \( |f(x)| \) is all non-negative numbers from 0 up to and including 1. In interval notation, this is: \[ [0, 1] \]