Math Problem Statement

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}

Solution

The problem asks for the domain of definition of the function f(x)f(x), where f(x)=maximum{x,1,x2}f(x) = \text{maximum} \{x, 1, x^2\} for all xRx \in \mathbb{R}.

The options provided are:

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

Solution:

To determine the domain of f(x)f(x), we analyze the behavior of f(x)=maximum{x,1,x2}f(x) = \text{maximum} \{x, 1, x^2\}:

  1. x2x^2 is always non-negative for all real xx, i.e., x20x^2 \geq 0.
  2. The value of maximum{x,1,x2}\text{maximum} \{x, 1, x^2\} depends on the relative magnitudes of xx, x2x^2, and 11. 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,x2}\{x, 1, x^2\} are equal:

  • x=x2x = x^2: Solve x2x=0x^2 - x = 0x(x1)=0x(x - 1) = 0, so x=0x = 0 or x=1x = 1.
  • x=1x = 1: Already included.
  • x=1x = -1: Not directly problematic because maximum\text{maximum} is well-defined.

However, all elements in R\mathbb{R} satisfy the function. The most plausible correction ##Upon reviewing the problem in detail, we conclude that the domain of f(x)f(x) depends on ensuring the maximum function f(x)=maximum{x,1,x2}f(x) = \text{maximum} \{x, 1, x^2\} has no undefined values.

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

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

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Math Problem Analysis

Mathematical Concepts

Functions
Set Theory
Analysis of Maximum Function

Formulas

f(x) = maximum {x, 1, x^2}

Theorems

Properties of maximum function

Suitable Grade Level

Grades 11-12