For what values of a and b isg(x) = { ax + 2b, x ≤ 0x² + 3a – b, 0 2continuous at every x?

To ensure the function g(x) is continuous at every x, we need to check the continuity at the points where the piecewise function changes, specifically at x = 0 and x = 2. For continuity at x = 0, the left-hand limit as x approaches 0 from the left (x ≤ 0) must equal the right-hand limit as x approaches 0 from the right (0 2), and both must equal g(2). Set x² + 3a - b = 3x - 5 at x = 2. Substitute x = 2 into both expressions: x² + 3a - b becomes 4 + 3a - b, and 3x - 5 becomes 6 - 5. Set 4 + 3a - b = 1 to find another relationship between a and b. Solve the system of equations from steps 3 and 5 to find the values of a and b that make g(x) continuous at every x.

Ch. 2 - Limits and Continuity

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Hass 15th Edition

Ch. 2 - Limits and Continuity

문제 2.5.50

2장, 문제 2.5.50

For what values of a and b is
g(x) = { ax + 2b, x ≤ 0
x² + 3a – b, 0 < x ≤ 2
3x – 5, x > 2
continuous at every x?

검증된 단계별 안내

To ensure the function g(x) is continuous at every x, we need to check the continuity at the points where the piecewise function changes, specifically at x = 0 and x = 2.

For continuity at x = 0, the left-hand limit as x approaches 0 from the left (x ≤ 0) must equal the right-hand limit as x approaches 0 from the right (0 < x ≤ 2), and both must equal g(0). Set ax + 2b = x² + 3a - b at x = 0.

Substitute x = 0 into both expressions: ax + 2b becomes 2b, and x² + 3a - b becomes 3a - b. Set 2b = 3a - b to find a relationship between a and b.

For continuity at x = 2, the right-hand limit as x approaches 2 from the left (0 < x ≤ 2) must equal the left-hand limit as x approaches 2 from the right (x > 2), and both must equal g(2). Set x² + 3a - b = 3x - 5 at x = 2.

Substitute x = 2 into both expressions: x² + 3a - b becomes 4 + 3a - b, and 3x - 5 becomes 6 - 5. Set 4 + 3a - b = 1 to find another relationship between a and b. Solve the system of equations from steps 3 and 5 to find the values of a and b that make g(x) continuous at every x.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point from both sides equals the function's value at that point. For piecewise functions, this means ensuring that the function values match at the boundaries of the defined intervals.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. In this case, g(x) has three different expressions depending on whether x is less than or equal to 0, between 0 and 2, or greater than 2. Analyzing continuity requires checking the transitions between these pieces.

Limit and Value Matching

To ensure continuity at the boundaries of the piecewise function, the limits from the left and right must equal the function's value at those points. This involves setting the expressions equal to each other at the transition points (x = 0 and x = 2) and solving for the parameters a and b.

For what values of a and b isg(x) = { ax + 2b, x ≤ 0x² + 3a – b, 0 < x ≤ 23x – 5, x > 2continuous at every x?