Video Transcript
What is the domain of the function
𝑓 of 𝑥 equals the square root of one minus 𝑥?
Recall that the domain of a
function is the set of all possible values of 𝑥 such that 𝑓 of 𝑥 is valid. By “valid,” we mean that the
function is well defined for that value and gives us a real number as an output. For example, if our function was
one over 𝑥, then zero would not be in the domain of the function because we cannot
divide one by zero. Since the function in question
involves a square root, the easiest way to approach this problem is to start off by
considering what inputs are valid for the square root function. By this, we mean 𝑓 of 𝑥 equals
the square root of 𝑥.
Since we are only considering real
numbers, the square root function is not defined for negative numbers. This means the domain is restricted
to values of 𝑥 greater than or equal to zero. We note that this is a greater than
or equal sign and not a strictly greater sign because 𝑥 equals zero is a perfectly
valid input. Another way to think of this is
that whatever expression we put under the square root has to be nonnegative. So, for example, whether we put two
𝑥, one over 𝑥, or, indeed, one minus 𝑥 into the function, we would need to check
whether that expression is greater than or equal to zero.
Returning to the original question,
we are considering the function that is the square root of one minus 𝑥. Therefore, we need to make sure the
expression inside the square root is nonnegative. This means we need to consider the
values of 𝑥 for which one minus 𝑥 is greater than or equal to zero. We can add 𝑥 to both sides of this
inequality to get one greater than or equal to 𝑥. And by swapping the sides around,
this is equivalent to 𝑥 is less than or equal to one. We note that it is also possible to
express this as an interval, as the values of 𝑥 from one to ∞. However, we will choose to keep it
as an inequality. Thus, the domain is the values of
𝑥 less than or equal to one.
