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20. Let $f(x)=\left\{\begin{array}{cl}a-x^{2} & \text { if } x \leq-1 \\ x-b & \...
Feb 1, 2024
20. Let f(x)={a−x2x−bamp; if x≤−1amp; if xgt;−1.
a) Determine the values of a and b that make the function discontinuous.
b) Graph the function.
c) Determine the value of each of the following, if it exists.
i) limx→−1+f(x)
ii) limx→−1−f(x)
iii) limx→−1f(x)
iv) f(−1)
Solution by Steps
step 1
To determine the values of a and b that make f(x) discontinuous, we need to find values that do not satisfy the continuity condition at x=−1
step 2
For x≤−1, f(x)=a−x2. For x > -1, f(x)=x−b
step 3
The function f(x) is continuous at x=−1 if the left-hand limit as x approaches −1 equals the right-hand limit as x approaches −1 and both are equal to f(−1)
step 4
The left-hand limit as x approaches −1 is a−(−1)2=a−1
step 5
The right-hand limit as x approaches −1 is −1−b
step 6
The function value at x=−1 is f(−1)=a−(−1)2=a−1
step 7
For f(x) to be discontinuous at x=−1, we need a−1=−1−b
step 8
Therefore, the function is discontinuous for any a and b such that a−1=−1−b
Answer
The function is discontinuous for any a and b such that a−1=−1−b.
Key Concept
Continuity at a Point
Explanation
A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are all equal. If any of these conditions are not met, the function is discontinuous at that point.
Solution by Steps
step 1
To graph the function f(x), we need to plot two separate pieces, one for x≤−1 and one for x > -1
step 2
For x≤−1, graph the parabola y=a−x2
step 3
For x > -1, graph the line y=x−b
step 4
The point of discontinuity, if any, will be at x=−1
Answer
The graph consists of a parabola for x≤−1 and a line for x > -1, with a potential point of discontinuity at x=−1.
Key Concept
Piecewise Function Graphing
Explanation
A piecewise function is graphed by plotting each piece within its respective domain and checking for continuity at the boundaries.
Solution by Steps
step 1
To find limx→−1+f(x), we consider the expression for f(x) when x > -1
step 2
The right-hand limit as x approaches −1 from the right is −1−b
Answer
limx→−1+f(x)=−1−b
Key Concept
Right-Hand Limit
Explanation
The right-hand limit of a function as x approaches a value is found by considering the function's behavior as x approaches that value from the right.
Solution by Steps
step 1
To find limx→−1−f(x), we consider the expression for f(x) when x≤−1
step 2
The left-hand limit as x approaches −1 from the left is a−1
Answer
limx→−1−f(x)=a−1
Key Concept
Left-Hand Limit
Explanation
The left-hand limit of a function as x approaches a value is found by considering the function's behavior as x approaches that value from the left.
Solution by Steps
step 1
To find limx→−1f(x), we need to compare the left-hand and right-hand limits at x=−1
step 2
If the left-hand limit a−1 equals the right-hand limit −1−b, then the limit exists and is equal to this common value
step 3
If a−1=−1−b, then the limit does not exist
Answer
limx→−1f(x) exists and is equal to a−1 if a−1=−1−b. Otherwise, the limit does not exist.
Key Concept
Limit of a Function at a Point
Explanation
The limit of a function at a point exists if and only if the left-hand and right-hand limits at that point are equal.
Solution by Steps
step 1
To find f(−1), we use the expression for f(x) when x≤−1
step 2
The function value at x=−1 is f(−1)=a−(−1)2=a−1
Answer
f(−1)=a−1
Key Concept
Function Value at a Point
Explanation
The value of a function at a point is determined by substituting the point into the function's expression that corresponds to the domain containing that point.
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