https://file.aiquickdraw.com/mathbot/file/17268575931497r0385ww.jpg

The problem in the image is asking for the graph of the piecewise function \( h(x) \) defined as follows:

\[
h(x) =
\begin{cases} 
x^3 - 1 & \text{if } x < 1 \\
0 & \text{if } 1 \leq x \leq 2 \\
\frac{1}{x} & \text{if } x > 2 
\end{cases}
\]

We also need to identify the domain and range of the function.

### Steps to approach this:

#### 1. **Graphing the function**:
- For \( x < 1 \), the function is \( h(x) = x^3 - 1 \), which is a cubic function shifted down by 1. It will smoothly decrease for negative \( x \) and approach \( -1 \) as \( x \to 1^- \).
  
- For \( 1 \leq x \leq 2 \), the function is a constant: \( h(x) = 0 \), so this will be a flat horizontal line between \( x = 1 \) and \( x = 2 \), inclusive.
  
- For \( x > 2 \), the function is \( h(x) = \frac{1}{x} \), which is a hyperbola. As \( x \to \infty \), \( h(x) \to 0 \) and as \( x \to 2^+ \), \( h(x) \) will approach \( \frac{1}{2} \).

#### 2. **Domain of the function**:
The domain is the set of all \( x \)-values where the function is defined. There are no restrictions on the value of \( x \), so the domain is:
\[
\text{Domain: } (-\infty, \infty)
\]

#### 3. **Range of the function**:
The range is the set of all possible output values of \( h(x) \).
- For \( x < 1 \), \( x^3 - 1 \) can take values from \( -\infty \) to \( 0 \) (but not including \( 0 \) because \( h(1) = 0 \)).
- For \( 1 \leq x \leq 2 \), \( h(x) = 0 \), so \( 0 \) is included.
- For \( x > 2 \), \( \frac{1}{x} \) approaches 0 from positive values, so \( h(x) \) will take values in \( (0, \frac{1}{2}] \).

Thus, the range is:
\[
\text{Range: } (-\infty, 0] \cup (0, \frac{1}{2}]
\]

Let me know if you'd like me to sketch the graph as well! Would you like any further clarification or details?

### Related Questions:
1. What is the behavior of the cubic function \( x^3 - 1 \) as \( x \to -\infty \)?
2. How would the graph of the function change if the condition \( 1 \leq x \leq 2 \) was altered to \( 1 < x < 2 \)?
3. What happens to the function \( \frac{1}{x} \) as \( x \to 2^+ \) and as \( x \to \infty \)?
4. Could you find any discontinuities in the function \( h(x) \)? If so, where?
5. How would you modify the function to ensure it is continuous over all \( x \)?

### Tip:
When sketching piecewise functions, always consider the limits at the breakpoints to determine if there are any jumps or discontinuities.

Sketch the graph of the function h(x), where h(x) is defined piecewise as: 
h(x) =
  x^3 - 1 if x < 1,
  0 if 1 <= x <= 2,
  1/x if x > 2.
Identify its domain and range.

Explore how to sketch the graph of the piecewise function h(x) with different expressions for various intervals of x. Learn to determine its domain and range, covering concepts such as cubic functions, constants, and hyperbolas. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation