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: 9780137616077Not the one you use?Change textbook

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Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.5.6a

Chapter 2, Problem 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.
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a. Does f (1) exist?

Verified step by step guidance

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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