Video Transcript
Determine the domain of the
function 𝑓 of 𝑥 equals the square root of negative eight 𝑥 plus 16.
We recall that the domain of a
function is the set of all possible values of 𝑥 such that 𝑓 of 𝑥 is defined. Considering we have been given a
square root function, we need to recall the associated restriction on the domain of
this type of function. A quick sketch of the familiar
square root function may help us to remember the restrictions on the domain.
Although the graph of the square
root function appears to become flatter for larger values of 𝑥, it continues to
increase without a bound. As indicated by the graph, the
domain of the main square root function is the set of all real, nonnegative
numbers. This domain can also be written as
the left-closed, right-open interval from zero to ∞, or as the inequality 𝑥 is
greater than or equal to zero.
More generally, the domain of the
composite function square root of 𝑔 of 𝑥 can be identified by finding the values
of 𝑥 satisfying 𝑔 of 𝑥 greater than or equal to zero. The function we have been given can
be written as 𝑓 of 𝑥 equals the square root of 𝑔 of 𝑥 plus 16, where 𝑔 of 𝑥 is
the linear expression negative eight 𝑥.
Then, according to the general
method for finding the domain of a composite square root function, we solve the
inequality 𝑔 of 𝑥 is greater than or equal to zero. To isolate the 𝑥, we first divide
both sides by negative eight. We recall that multiplying or
dividing an inequality by a negative changes the direction of the inequality. Therefore, 𝑥 is less than or equal
to zero.
We must still consider whether or
not the plus 16 has any effect on the domain of the function. When a real number is added outside
of the square root function, this indicates a vertical shift. In this case, we have a vertical
shift up 16, which changes the 𝑦-coordinates but has no effect on the
𝑥-coordinates and, thus, no effect on the domain. Therefore, the only restriction on
the domain is that 𝑥 must be less than or equal to zero.
In conclusion, the domain of the
function 𝑓 of 𝑥 equals the square root of negative eight 𝑥 plus 16 is the set of
real numbers less than or equal to zero. We can also write our domain as the
left-open, right-closed interval from negative ∞ to zero.
