Final answer:
The range of the given function, u(x) = (1)/(x-4)² - 2, is [-2, ∞ ) . This includes all values from -2 (inclusive) to positive infinity (exclusive).
Step-by-step explanation:
The function given is u(x) = (1)/(x-4)² - 2. To find the range of this function, we need to consider the values that u(x) can take, as 'x' varies over the domain. As we can see, this is a rational function where the denominator, (x-4)², cannot be 0 (so x cannot be 4), and the function will be defined for all other values of x.
Moving to the function, we know that square of any real number is always non-negative - that is, (x-4)² is always greater or equal to 0. Dividing '1' by a positive value, we'll get some positive number or at the very least 0 (approaching infinity at the boundary). So, 1 / (x-4)² will always be greater or equal to 0. Subtracting '2' will shift the function 2 units down on y-axis. Hence, the function can take any value from -2 (inclusive) till positive infinity but it cannot be above -2.
So, the range of the function u(x) = (1)/(x-4)² - 2 is [ -2, ∞ ) in interval notation, where the square bracket means the number -2 is included in the range, and the parenthesis means positive infinity is not included.
Learn more about Range of a function