find the points of discontinuity of the given function f(x) =x+3/|x2+3x|
the denominator is zero at x = 0 and at x = -3
try sketching a graph of it right around x = 0 and around x = -3
To find the points of discontinuity of the function f(x) = (x+3) / |x^2+3x|, we need to identify the values of x where the function is undefined.
The function will be undefined if the denominator, |x^2+3x|, equals zero. In other words, we need to find the values of x that make the expression inside the absolute value equal to zero.
|x^2+3x| = 0
To solve this equation, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: x^2+3x > 0 (positive expression)
For x^2+3x to be positive, we can consider two subcases:
Subcase 1.1: x^2+3x > 0 when x > 0
In this case, we have a quadratic expression that is always positive. Therefore, there are no values of x that make x^2+3x greater than zero.
Subcase 1.2: x^2+3x > 0 when x < 0
We have a quadratic expression that is negative when -3 < x < 0. However, a negative value inside the absolute value would still yield a positive value. So, there are no values of x that make x^2+3x greater than zero in this subcase.
Case 2: x^2+3x < 0 (negative expression)
For x^2+3x to be negative, we can consider two subcases:
Subcase 2.1: x^2+3x < 0 when x > 0
We have a quadratic expression that is positive for x > 0. However, a positive value inside the absolute value would still yield a positive value. So, there are no values of x that make x^2+3x less than zero in this subcase.
Subcase 2.2: x^2+3x < 0 when x < 0
In this subcase, x^2+3x is negative when -3 < x < 0. A negative value in the denominator would make the function undefined. Therefore, the function is undefined for x in the interval (-3, 0).
Combining the results from both cases, we have found that the function f(x) = (x+3) / |x^2+3x| is undefined for x in the interval (-3, 0). These are the points of discontinuity of the function.