Exercises 5–10 refer to the functionf(x) = { x² − 1, −1 ≤ x a. Does f (1) exist?

Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Hass 15th Edition

Ch. 2 - Limits and Continuity

문제 2.5.6a

2장, 문제 2.5.6a

Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
<IMAGE>
a. Does f (1) exist?

검증된 단계별 안내

Identify the piecewise function definition for f(x) at x = 1. According to the problem, f(x) is defined as 1 when x = 1.

To determine if f(1) exists, check if there is a specific value assigned to f(x) at x = 1 in the piecewise function.

Since the piecewise function explicitly defines f(x) = 1 for x = 1, f(1) exists and is equal to 1.

In general, for a function value to exist at a specific point, the function must be defined at that point in the piecewise definition.

Conclude that f(1) exists because the piecewise function provides a specific value for f(x) at x = 1.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has distinct formulas for various intervals of x, which means that the output depends on which interval the input falls into. Understanding how to evaluate piecewise functions is crucial for determining function values at specific points.

Function Existence

For a function to exist at a certain point, it must have a defined output for that input. This involves checking if the input falls within the domain of the function and if the corresponding expression yields a real number. In the context of the question, we need to evaluate f(1) to see if it is defined according to the piecewise conditions.

Continuity at a Point

Continuity at a point means that the function's value at that point matches the limit of the function as it approaches that point from both sides. While the question specifically asks about the existence of f(1), understanding continuity helps in analyzing the behavior of the function around that point, which can be important for further questions regarding limits or derivatives.

Exercises 5–10 refer to the functionf(x) = { x² − 1, −1 ≤ x < 02x, 0 < x < 11, x = 1−2x + 4, 1 < x < 20, 2 < x < 3graphed in the accompanying figure.<IMAGE>a. Does f (1) exist?