Find the domain and range of the function                1 1 0 1 1 1 1 ( ) x if x if x x if x f x (2 Marks)

Let's analyze the piecewise function given:

\[
f(x) = \begin{cases} 
-x - 1 & \text{if } x < -1 \\
x^2 & \text{if } -1 \leq x \leq 1 \\
x + 1 & \text{if } x > 1 
\end{cases}
\]

### Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. 

- For the first piece, \(-x - 1\), there is no restriction on \(x\) other than \(x < -1\).
- For the second piece, \(x^2\), there is no restriction on \(x\) other than \(-1 \leq x \leq 1\).
- For the third piece, \(x + 1\), there is no restriction on \(x\) other than \(x > 1\).

Since these three pieces cover all real numbers, the domain of \(f(x)\) is all real numbers:

\[
\text{Domain: } (-\infty, \infty)
\]

### Range:
The range of a function is the set of all possible output values (y-values).

- For \(x < -1\), \(f(x) = -x - 1\). As \(x\) approaches \(-1\) from the left, \(f(x)\) approaches 0. As \(x\) decreases towards \(-\infty\), \(f(x)\) increases towards \(\infty\). Thus, the range for this piece is \((-\infty, 0)\).
- For \(-1 \leq x \leq 1\), \(f(x) = x^2\). The minimum value occurs at \(x = 0\), giving \(f(x) = 0\). The maximum value occurs at \(x = \pm1\), giving \(f(x) = 1\). Thus, the range for this piece is \([0, 1]\).
- For \(x > 1\), \(f(x) = x + 1\). As \(x\) increases from 1, \(f(x)\) increases from 2 onwards. Thus, the range for this piece is \((2, \infty)\).

By combining the ranges of the three pieces, we get the overall range:

\[
\text{Range: } (-\infty, 0) \cup [0, 1] \cup (2, \infty)
\]

### Summary:
- **Domain:** \((-∞, ∞)\)
- **Range:** \((-∞, 0) \cup [0, 1] \cup (2, ∞)\)

Would you like more details or have any questions?

### Related Questions:
1. How do you graph piecewise functions?
2. What are the different methods to find the range of a function?
3. How does the domain of a function differ from its range?
4. Can a function have a discontinuous range?
5. What are the implications of a function being defined as piecewise?
6. How do you handle inequalities when determining the domain and range?
7. Can a piecewise function be continuous?
8. How can the inverse of a piecewise function be determined?

### Tip:
When determining the range of a piecewise function, it's helpful to consider the range of each piece separately and then combine them, checking for any overlaps or gaps.

Discover how to determine the domain and range of a piecewise function f(x) = {-x - 1 if x < -1, x^2 if -1 ≤ x ≤ 1, x + 1 if x > 1}. Understand the mathematical concepts involved and learn how to analyze piecewise functions effectively. Suitable for Grades 11-12 students studying advanced functions.

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