The domain of a function built from fractions, square roots and logarithms, in interval notation. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
18 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
At x = -2 the radicand is 0 (it must be ≥ 0).✓ Proved
\[ \left. x - 3 \right|_{\substack{ x=3 }} = 0 \]
At x = 3 the denominator is 0.✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( \left[-2, 3) \cup (3, \infty) \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the domain of \( \displaystyle f(x) = \frac{\sqrt{x - 2}}{x - 3} \). Give it in interval notation.
\[ \left. x - 2 \right|_{\substack{ x=2 }} = 0 \]
At x = 2 the radicand is 0 (it must be ≥ 0).✓ Proved
\[ \left. x - 3 \right|_{\substack{ x=3 }} = 0 \]
At x = 3 the denominator is 0.✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( \left[2, 3) \cup (3, \infty) \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( \left[-2, -1) \cup (-1, \infty) \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
At x = -5 the log's argument is 0 (it must be > 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (-5, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
At x = -4 the radicand is 0 (it must be ≥ 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (-\infty, -4\right] \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
At x = -1 the log's argument is 0 (it must be > 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (-1, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
At x = -4 the radicand is 0 (it must be ≥ 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( \left[-4, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the domain of \( \displaystyle f(x) = \ln{\left(x - 3 \right)} + 3 \). Give it in interval notation.
\[ \left. x - 3 \right|_{\substack{ x=3 }} = 0 \]
At x = 3 the log's argument is 0 (it must be > 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (3, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
At x = 1 the radicand is 0 (it must be ≥ 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( \left[1, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the domain of \( \displaystyle f(x) = \ln{\left(x - 4 \right)} - 3 \). Give it in interval notation.
\[ \left. x - 4 \right|_{\substack{ x=4 }} = 0 \]
At x = 4 the log's argument is 0 (it must be > 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (4, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
At x = -3 the radicand is 0 (it must be ≥ 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (-\infty, -3\right] \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the domain of \( \displaystyle f(x) = \ln{\left(x - 3 \right)} \). Give it in interval notation.
\[ \left. x - 3 \right|_{\substack{ x=3 }} = 0 \]
At x = 3 the log's argument is 0 (it must be > 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (3, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
At x = -1 the log's argument is 0 (it must be > 0).✓ Proved
Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.
Reviewed
Answer \( (-1, \infty) \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify