Determining the Domains of Expressions with Radicals (2.13.3)

Question 1

• The domain is the set of all values that can be used for x in an expression.

Answer 1

• The domain is the set of all values that can be used for x in an expression.

Question 2

• The radicand is the value or expression under the radical sign. It must be positive in order to have a real number root.

Answer 2

• The radicand is the value or expression under the radical sign. It must be positive in order to have a real number root.

Question 3

• Solving inequalities:
• Factor where possible.
• Determine the points where each factor equals zero.
• Determine what intervals produce results that satisfy the inequality as originally stated.

Answer 3

• Solving inequalities:
• Factor where possible.
• Determine the points where each factor equals zero.
• Determine what intervals produce results that satisfy the inequality as originally stated.

Question 4

In this type of problem, find out what values of x will help
out.
x
2
– 16 must be positive or there is no real number square
root. That fact leads to setting up an inequality to solve for
x.
Resort to the usual inequality tricks:
1. Factor to get (x + 4) and (x - 4).
2. Set each factor equal to 0.
3. Solve for x .
4. Mark those values on a number line.
5. Choose random points from each interval to determine
which of them satisfy the inequality:
–5 from the left: produces +9, >0; it works.
0 from the middle: produces –16, 0; it works.
Mark the left and right intervals as solutions on the number
line and include both –4 and +4 because the expression equals
0 at those points.
What this means is that values between –4 and +4 have been
excluded from the domain of x because they create a negative
value under the radical.

Answer 4

In this type of problem, find out what values of x will help
out.
x
2
– 16 must be positive or there is no real number square
root. That fact leads to setting up an inequality to solve for
x.
Resort to the usual inequality tricks:
1. Factor to get (x + 4) and (x - 4).
2. Set each factor equal to 0.
3. Solve for x .
4. Mark those values on a number line.
5. Choose random points from each interval to determine
which of them satisfy the inequality:
–5 from the left: produces +9, >0; it works.
0 from the middle: produces –16, 0; it works.
Mark the left and right intervals as solutions on the number
line and include both –4 and +4 because the expression equals
0 at those points.
What this means is that values between –4 and +4 have been
excluded from the domain of x because they create a negative
value under the radical.