Question

Алгебра. Помогите, плиз Найдите область определения функции y=lg⁡〖(2x-3)/(x+7)〗

Answer 1

У=lg (2х-3)/(х+7)
(2х-3)/(х+7)>0    x+7≠0 ⇒ x≠-7

1{   2x-3>0  и  x+7>0  х>1,5 и х>-7    x∈ (1,5;+∞)

2{    2x-3<0  и  x+7<0  х<1,5 и х<-7   x∈(-∞;-7)

Ответ: x∈ (-∞;-7)∪(1,5;+∞)

Answer 2

Finding the Domain of the Function y = lg⁡((2x-3)/(x+7))

To find the domain of the function y = lg⁡((2x-3)/(x+7)), we need to consider the values of x for which the function is defined. The domain of a logarithmic function is the set of all positive real numbers for which the argument of the logarithm is positive.

The argument of the logarithm in this case is (2x-3)/(x+7). To find the domain, we need to ensure that the argument is positive.

The domain of the function y = lg⁡((2x-3)/(x+7)) can be found by solving the inequality (2x-3)/(x+7) > 0.

Solution:

To solve the inequality (2x-3)/(x+7) > 0, we can use the method of intervals. We need to find the intervals of x for which the expression (2x-3)/(x+7) is positive.

1. Find the critical points by setting the numerator and denominator equal to zero and solving for x. 2. Use the critical points to create intervals and test a value from each interval in the inequality to determine where it is true.

Let's proceed with finding the critical points and determining the intervals where the inequality is satisfied.

Critical Points and Intervals:

The critical points are the values of x that make the expression (2x-3)/(x+7) equal to zero or undefined.

The critical points are: - x = 3/2 (from the numerator 2x-3 = 0) - x = -7 (from the denominator x+7 = 0)

We have two critical points: x = 3/2 and x = -7. These critical points divide the real number line into three intervals: (-∞, -7), (-7, 3/2), and (3/2, ∞).

Testing Intervals:

Now, we need to test a value from each interval in the inequality (2x-3)/(x+7) > 0 to determine where it is true.

1. For the interval (-∞, -7): - Let's choose x = -8. Substituting x = -8 into the inequality gives us (-19)/(-1) > 0, which is true.

2. For the interval (-7, 3/2): - Let's choose x = 0. Substituting x = 0 into the inequality gives us (-3)/(7) < 0, which is false.

3. For the interval (3/2, ∞): - Let's choose x = 2. Substituting x = 2 into the inequality gives us (1)/(9) > 0, which is true.

Conclusion:

Based on the testing of intervals, we have found that the inequality (2x-3)/(x+7) > 0 is true for the intervals (-∞, -7) and (3/2, ∞).

Therefore, the domain of the function y = lg⁡((2x-3)/(x+7)) is x < -7 or x > 3/2. These are the values of x for which the function is defined.