Question Video: Finding the Domain of Rational Functions | Nagwa Question Video: Finding the Domain of Rational Functions | Nagwa

# Question Video: Finding the Domain of Rational Functions Mathematics • Second Year of Secondary School

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Determine the domain of the function 𝑓(𝑥) = 2/√(86 − |𝑥|).

03:09

### 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.

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