Question Video: Finding the Domain and Range of a Function Involving Nested Square Roots and Cube Roots Mathematics

Video Transcript

Consider the function 𝑓 of 𝑥 equals the cubed root of 125 minus the square root of two 𝑥 plus three. Part (a): Find the domain of 𝑓 of 𝑥. Part (b): Find the range of 𝑓 of 𝑥.

We have here a cube root function and then a square root function in the expression within the cube root. We recall first that the domain of a cube root function is the set of all real numbers. So as there is no restriction on the domain of the cube root function, we only need to consider the restriction for the square root function.

The domain of a square root function must be nonnegative. So we have the inequality two 𝑥 plus three is greater than or equal to zero. Solving this inequality leads to 𝑥 is greater than or equal to negative three over two. So we find that the domain of the function 𝑓 of 𝑥 is the left-closed, right-open interval from negative 1.5 to ∞, because we can only evaluate the entire expression under the cube root when 𝑥 is in this interval.

Let’s now consider the range of 𝑓 of 𝑥. We can write our function as 𝑓 of 𝑥 equals the cube root of 125 minus 𝑎, where 𝑎 is equal to the square root of two 𝑥 plus three. We know that the range of a square root function is the set of all nonnegative values. And so 𝑎 is greater than or equal to zero. The largest value of 𝑓 of 𝑥 will correspond to the smallest value of 𝑎, which is zero. So the largest value of 𝑓 of 𝑥 is the cube root of 125 minus zero, which is the cube root of 125, which is five.

The smallest value of 𝑓 of 𝑥 will correspond to the largest value of 𝑎. So as 𝑎 tends to ∞, 𝑓 of 𝑥 will tend to the cube root of negative ∞, which itself tends to negative ∞. As 𝑓 of 𝑥 is a continuous function, its range will therefore include all the values between its smallest and largest value, which is the left-open, right-closed interval from negative ∞ to five.

So we’ve completed the problem and found the domain and range of this fairly complicated radical function, which involves both square and cube roots. The domain is the left-closed, right-open interval from negative 1.5 to ∞. And the range is the right-open, left-closed interval from negative ∞ to five.