Let f(x)= sqrt((x/x+3)-5). Find the domain.
you know that x/(x+3) is undefined at x = -3
since √n is defined only for n >= 0, we need
x/(x+3)-5 >= 0
x/(x+3) > 5
x > 5x+15
-15 > 4x
-15/4 > x
So, the domain is all reals less than -15/4. Note that this automatically excludes x = -3.
To find the domain of a function, we need to consider the values of x for which the function is defined. In this case, we have the function f(x) = sqrt((x/x+3) - 5).
The square root function is defined only for non-negative inputs; therefore, the expression inside the square root must be greater than or equal to zero.
Setting the expression inside the square root to be greater than or equal to zero:
(x/x + 3) - 5 ≥ 0
Now, let's solve this inequality step by step:
x/(x + 3) - 5 ≥ 0
To combine the fractions, we need a common denominator, which is (x + 3).
Multiply the first fraction by (x + 3)/(x + 3):
(x(x + 3)/(x + 3) - 5(x + 3))/(x + 3) ≥ 0
Simplifying the expression:
(x^2 + 3x - 5x - 15)/(x + 3) ≥ 0
(x^2 - 2x - 15)/(x + 3) ≥ 0
Now, let's factor the numerator:
(x + 3)(x - 5)/(x + 3) ≥ 0
(x - 5) ≥ 0
Now, we have the inequality:
x - 5 ≥ 0
Solving for x:
x ≥ 5
Therefore, the domain of the function f(x) = sqrt((x/x + 3) - 5) is x ≥ 5.