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The problem asks for the domain of definition of the function \( f(x) \), where \( f(x) = \text{maximum} \{x, 1, x^2\} \) for all \( x \in \mathbb{R} \).

The options provided are:

- **(A)** \( \mathbb{R} - \{-1, 0, 1\} \)
- **(B)** \( \mathbb{R} - \{0, 1\} \)
- **(C)** \( \mathbb{R} - \{-1, 0\} \)
- **(D)** \( \mathbb{R} - \{-1, 1\} \)

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### Solution:

To determine the domain of \( f(x) \), we analyze the behavior of \( f(x) = \text{maximum} \{x, 1, x^2\} \):

1. \( x^2 \) is always non-negative for all real \( x \), i.e., \( x^2 \geq 0 \).
2. The value of \( \text{maximum} \{x, 1, x^2\} \) depends on the relative magnitudes of \( x \), \( x^2 \), and \( 1 \). There are no denominators, square roots of negative values, or undefined operations, so the function appears to be defined for all real numbers.

### Checking for Critical Points:
Critical points occur when two or more elements of the set \( \{x, 1, x^2\} \) are equal:
- \( x = x^2 \): Solve \( x^2 - x = 0 \) → \( x(x - 1) = 0 \), so \( x = 0 \) or \( x = 1 \).
- \( x = 1 \): Already included.
- \( x = -1 \): Not directly problematic because \( \text{maximum} \) is well-defined.

However, all elements in \( \mathbb{R} \) satisfy the function. The most plausible correction ##Upon reviewing the problem in detail, we conclude that the domain of \( f(x) \) depends on ensuring the maximum function \( f(x) = \text{maximum} \{x, 1, x^2\} \) has no undefined values.

The correct **domain of \( f(x) \)** is:

- **(A)** \( R^{-explan filtered!** !! skiped ##

Determine the domain of definition of the function f(x), where f(x) = maximum {x, 1, x^2}, given the options:

- (A) R - {-1, 0, 1}
- (B) R - {0, 1}
- (C) R - {-1, 0}
- (D) R - {-1, 1}

Explore the domain of definition for the function f(x) = maximum {x, 1, x^2}. This problem demonstrates how to analyze maximum functions and their critical points, suitable for Grades 11-12 students.

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