https://file.aiquickdraw.com/mathbot/file/1727673829616zvntcp4f.jpg

Using the given graph of the function​ f, find the following.

​(a) the​ intercepts, if any

​(b) its domain and range

​(c) the intervals on which it is​ increasing, decreasing, or constant

​(d) whether it is​ even, odd, or neither

Let’s analyze the graph and answer each of the following questions.

### (a) Intercepts
- **x-intercepts**: Points where the function crosses the x-axis. From the graph, the function crosses the x-axis at \(x = -2\) and \(x = 2\). So the x-intercepts are \((-2, 0)\) and \((2, 0)\).
- **y-intercept**: The point where the function crosses the y-axis. In the graph, the function crosses the y-axis at \(y = 2\), so the y-intercept is \((0, 2)\).

### (b) Domain and Range
- **Domain**: The set of all x-values for which the function is defined. The graph spans from \(x = -4\) to \(x = 4\). Therefore, the domain is:
  \[
  \text{Domain: } [-4, 4]
  \]
- **Range**: The set of all y-values the function takes. From the graph, the lowest point is at \(y = 0\) and the highest point is \(y = 4\). Therefore, the range is:
  \[
  \text{Range: } [0, 4]
  \]

### (c) Intervals of Increase, Decrease, and Constant
- **Increasing**: The function is increasing where the slope of the line is positive. From the graph:
  - Increasing from \(x = -2\) to \(x = 0\).
  - Increasing from \(x = 2\) to \(x = 4\).

- **Decreasing**: The function is decreasing where the slope is negative. From the graph:
  - Decreasing from \(x = -4\) to \(x = -2\).
  - Decreasing from \(x = 0\) to \(x = 2\).

- **Constant**: There are no intervals where the function is constant since the graph consists entirely of sloped lines.

### (d) Symmetry: Even, Odd, or Neither
- **Even function**: A function is even if \(f(x) = f(-x)\) for all x in the domain, meaning the graph is symmetric about the y-axis.
- **Odd function**: A function is odd if \(f(-x) = -f(x)\), meaning the graph is symmetric about the origin.

From the graph, it is neither symmetric about the y-axis nor the origin. Hence, the function is **neither even nor odd**.

---

Would you like further clarification on any part? 

Here are five related questions to expand your understanding:
1. How would the graph change if the function were even or odd?
2. What is the significance of the x- and y-intercepts in real-world applications?
3. Can the domain of a function ever be larger than what is shown on the graph?
4. How do you determine whether a piecewise function is increasing or decreasing?
5. How do you find the symmetry of more complex functions analytically?

**Tip**: Always check both symmetry and behavior at intercepts when analyzing the nature of a function graph.

Using the given graph of the function​ f, find the following. (a) the intercepts, if any (b) its domain and range (c) the intervals on which it is​ increasing, decreasing, or constant (d) whether it is​ even, odd, or neither

Explore the graph of function f to identify its intercepts, domain, range, intervals of increase and decrease, and whether the function is even, odd, or neither. A complete solution is provided, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation