Question

Find All Values of x where the function is discontinuous, and from the left and from the right at any values of x=19.

f(x)={(1 if x<2),(x+3 if 2<=x<=4),(7 if x>4):}

Answer 1

To find the discontinuities of the function, we need to look at x=2 and x=4. The function is discontinuous at x=2 but continuous at x=4. Around x=19, the function is continuously equal to 7 from both the left and the right.

Step-by-step explanation:

To find all values of x where the function is discontinuous, we need to examine the given piecewise function:

Discontinuities can occur where the function transitions from one piece to another, which in this case is at x=2 and x=4. To determine if these are actual points of discontinuity we need to check the limits from the left and right at each point as well as the function's value at those points.

For x=2, the limit from the left of 2 is 1, and the limit from the right of 2 is <2+3>, which equals 5. Since these are not equal, there is a discontinuity at x=2. Similarly, for x=4, the limit from the left of 4 is <4+3>, which equals 7, and the limit from the right is the constant value 7. Here, the function is continuous.

Around x=19, the function is a constant 7 which means it is continuous at that point from the left and from the right.