Find the domain of y=x+3/4vx^2-9

Question

The answer of this question is (-∞,-5)U(-5,-3]U[3,5)U(5,∞). I don't know how the 5 and -5 come out.

Answer 1

The function as indicated in the last comment is:

y = [x+3] / [ 4 - √(x² - 9)]

The domain is determined by the denominator of that fraction. The expression under the radical (radicand) must be ≥ 0 (cannot take the square root of a negative number), AND the denominator cannot equal zero (cannot divide by zero).

x² - 9 ≥ 0 AND 4 - √(x² - 9) ≠ 0

When does the denominator equal zero:

4 = √(x² - 9)

16 = x² - 9

x² = 25

x = ±5 (since we squared both sides to solve this radical equation make sure neither is an extraneous root by substituting into the original equation).

x² - 9 ≥ 0

x² ≥ 9

|x| ≥ 3

x ≥ 3 OR x ≤ -3

Plotting those two rays on a number line, then removing -5 and 5 gives a picture of the solution set.

<------------(-5)-----[-3]xxxxxxxxx[3]----(5)-------------------->