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For the piecewise function example, verify the continuity at x = 2.
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At x = 2, f(2) = 8 in both functions, confirming it is continuous.
For the piecewise function f(x) = 5x + 3 (x < 1) and f(x) = x² + 4 (1 ≤ x < 2), determine the continuity at x = 1.
At x = 1, f(1) = 8 (from the first function) and f(1) = 5 (from the second function), indicating a discontinuity.
Identify the points of discontinuity for the function (3x + 2)/(x + 2)(x - 5) and classify them.
There is a vertical asymptote at x = 5 (infinite discontinuity) and a hole at x = -2 (removable discontinuity).
Determine the constants a and b for the following piecewise function: ax + 5 (x < 1) and x² - bx + 9 (1 ≤ x < 4).
For x = 1, a + b = 5. For x = 4, solving the equation gives a = 2 and substituting back gives b = 3.
Describe the discontinuity of the function |x|/x.
The function is undefined at x = 0 and shows a jump discontinuity.
What type of discontinuity is associated with vertical asymptotes?
Infinite discontinuity, which is non-removable, is associated with vertical asymptotes.
How do you find the constant c for the piecewise function cx + 3 (x < 2) and 3x + c (x ≥ 2) to ensure continuity at x = 2?
Set the expressions equal at x = 2 and solve: 2c + 3 = 6 + c, resulting in c = 3.
Explain the concept of setting equal values to find constants for continuous piecewise functions.
To ensure a piecewise function is continuous at a point where pieces join, set the expressions equal at that point and solve for the unknown constants.
How do you identify an infinite discontinuity on a graph?
An infinite discontinuity is identified by the graph approaching positive or negative infinity at a point, commonly linked with vertical asymptotes.
Where is the point of discontinuity for the function 5/(x + 2), and what type is it?
There is a vertical asymptote at x = -2, indicating an infinite discontinuity.
For the function 1/x², identify the point of discontinuity and its type.
There is a vertical asymptote at x = 0, indicating an infinite discontinuity.
Describe a removable discontinuity.
A removable discontinuity, or hole, occurs where the graph is undefined at a point, but can be defined by a limit.
What is the definition of a non-removable jump discontinuity?
A non-removable jump discontinuity occurs when the graph shows different left-hand and right-hand limits at a certain point.
Find the constant a for the function ax - 2 (x < 3) and x² - 5 (x ≥ 3) to ensure continuity at x = 3.
Set the expressions equal at x = 3 and solve: 3a - 2 = 9 - 5, resulting in a = 2.
What are the characteristics of a continuous graph?
A continuous graph has no jumps or breaks.
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