If domain of f(x) is [-1, 2], then find the domain of f([x]-x^2+4), where [ ]denotes greatest integer function.

To determine the domain of the function \( f([x] - x^2 + 4) \), given that the domain of \( f(x) \) is \([-1, 2]\), we first need to analyze the expression inside the function \( f \). The goal is to find the values of \( x \) for which \( [x] - x^2 + 4 \) falls within the interval \([-1, 2]\).

Understanding the Components

The expression \( [x] \) represents the greatest integer less than or equal to \( x \). This means that for any real number \( x \), \( [x] \) will be an integer. For example:

• If \( x = 1.5 \), then \( [x] = 1 \).
• If \( x = -0.3 \), then \( [x] = -1 \).

Setting Up the Inequality

We need to find the values of \( x \) such that:

-1 ≤ [x] - x² + 4 ≤ 2

Breaking Down the Inequalities

Let's break this down into two separate inequalities:

1. The Lower Bound

Starting with the first part:

-1 ≤ [x] - x² + 4

This simplifies to:

[x] - x² ≥ -5

Rearranging gives:

[x] ≥ x² - 5

2. The Upper Bound

Now, for the upper bound:

[x] - x² + 4 ≤ 2

This simplifies to:

[x] - x² ≤ -2

Rearranging gives:

[x] ≤ x² - 2

Finding the Domain

Now we need to find the values of \( x \) that satisfy both inequalities:

[x] ≥ x² - 5 and [x] ≤ x² - 2

Analyzing the Inequalities

Since \( [x] \) is an integer, we can denote \( [x] = n \), where \( n \) is an integer. Thus, we can rewrite the inequalities as:

• n ≥ x² - 5
• n ≤ x² - 2

Combining the Inequalities

From these inequalities, we can derive:

x² - 5 ≤ n ≤ x² - 2

This implies:

x² - 5 ≤ x² - 2

Which simplifies to:

-5 ≤ -2

This is always true, so we need to find the integer values of \( n \) that satisfy both conditions.

Finding Integer Values

Next, we can express \( n \) in terms of \( x \):

x² - 5 ≤ n ≤ x² - 2

For \( n \) to be an integer, we can set \( n = k \), where \( k \) is an integer. Thus, we have:

• k ≥ x² - 5
• k ≤ x² - 2

Finding the Range of x

Now, we can find the range of \( x \) for each integer \( k \). For example:

• If \( k = -1 \): -1 ≥ x² - 5 gives \( x² ≤ 4 \) or \( -2 ≤ x ≤ 2 \).
• If \( k = 0 \): 0 ≥ x² - 5 gives \( x² ≤ 5 \) or \( -√5 ≤ x ≤ √5 \).
• If \( k = 1 \): 1 ≥ x² - 5 gives \( x² ≤ 6 \) or \( -√6 ≤ x ≤ √6 \).

Final Domain Calculation

After analyzing these ranges, we can conclude that the domain of \( f([x] - x² + 4) \) is the set of all \( x \) values that satisfy the inequalities derived from the integer values of \( n \). Thus, the final domain is:

[-2, 2] since this is the intersection of all valid ranges derived from the integer values of \( n \).

In summary, the domain of \( f([x] - x² + 4) \) is \([-2, 2]\), which ensures that the expression inside \( f \) remains within the defined limits of the original function \( f(x) \).