Video Transcript
Determine the domain of the
function 𝑓 of 𝑥 equals two over the square root of 86 minus the absolute value of
𝑥.
We recall that the domain of any
function 𝑓 of 𝑥 is the set of all possible values of 𝑥 such that 𝑓 of 𝑥 is
defined. In the given function, we observe
two types of restrictions to the domain. First, we bring our attention to
the denominator. Since the denominator of a fraction
cannot equal zero, we need to exclude the case when the square root of 86 minus the
absolute value of 𝑥 equals zero. For the square root of an algebraic
expression to equal zero, the expression must equal zero. Therefore, our function 𝑓 of 𝑥
cannot be defined when 86 minus the square root of 𝑥 equals zero.
We also need to consider the
restrictions imposed by the square root. In particular, we recall that the
square root function cannot take a negative number. Thus, 86 minus the absolute value
of 𝑥 must be nonnegative, in other words, greater than or equal to zero. However, if we combine this with
our first restriction, 86 minus the absolute value of 𝑥 is strictly greater than
zero because we first found that it cannot equal zero.
To solve the inequality, we must
first isolate the absolute value on one side of the inequality. We begin by subtracting 86 from
each side of the inequality then divide each side by negative one. Whenever we multiply or divide an
inequality by a negative, the direction of the inequality changes. So we are left with the absolute
value of 𝑥 is strictly less than 86.
We recall that the absolute value
of 𝑥 can be interpreted as the distance from zero. This means we are looking for the
values of 𝑥 that are within 86 units from zero. We can picture these values of 𝑥
on a number line, between negative 86 and positive 86. We shade these values in orange,
leaving the interval open on the left and on the right. Thus, the values of the domain can
be written as the compound inequality negative 86 is less than 𝑥 is less than 86 or
in interval notation as the open interval between negative 86 and positive 86.
To find this answer, we restricted
the denominator to nonzero values and restricted the expression under the square
root to nonnegative values. Then, we solved the resulting
absolute value inequality for 𝑥. Therefore, the domain of 𝑓 of 𝑥
equal to two over the square root of 86 minus the absolute value of 𝑥 is the set of
real numbers between negative 86 and positive 86.
