Question

How many discontinuities does the following piecewise function have?

f(x) = {(3eˣ⁻³ + 1, x < -3), (2/3)x² - 1, -3 < x <= 3, -7/2x + 29/2, 3 < x < 5, log(2x-4), x >= 5}
Option 1: 1
Option 2: 2
Option 3: 3
Option 4: 4

Answer 1

The piecewise function has 1 discontinuity, which occurs at x = 5, since there is a mismatch in the function values from either side of this point.

Step-by-step explanation:

To determine how many discontinuities the piecewise function has, we need to examine each interval transition point to see if the function values match up from either side of the point:

• At x = -3, the transition from 3ex-3 + 1 to (2/3)x2 - 1 needs to be continuous.
• At x = 3, the transition from (2/3)x2 - 1 to -7/2x + 29/2 needs to be continuous.
• At x = 5, the transition from -7/2x + 29/2 to log(2x-4) needs to be continuous.

We need to check each interval's endpoints by substituting the boundary values into the corresponding functions:

1. When x = -3, we get 3e0 + 1 = 4 from the first function and (2/3)(-3)2 - 1 = 4 from the second function, so they match, and there is no discontinuity at x = -3.
2. For x = 3, the second function gives (2/3)(3)2 - 1 = 5 and the third function gives -7/2(3) + 29/2 = 5. There's no discontinuity at x = 3.
3. When x = 5, the third function yields -7/2(5) + 29/2 = 2, and the fourth function log(2(5)-4) = log(6) is not equal to 2, which means there is a discontinuity at x = 5.

Therefore, the piecewise function has 1 discontinuity.