https://file.aiquickdraw.com/mathbot/file/1727399093449thvt8cnf.jpg

Let's analyze and solve the given questions based on the provided graph:

### (a) State the value of \( f(1) \).
- From the graph, when \( x = 1 \), \( f(1) \) is clearly at 1.  
  **Answer**: \( f(1) = 1 \).

### (b) Estimate the value of \( f(-1) \).
- From the graph, when \( x = -1 \), the corresponding \( f(x) \) value appears to be approximately 0.5.
  **Answer**: \( f(-1) \approx 0.5 \).

### (c) For what values of \( x \) is \( f(x) = 1 \)?
- From the graph, \( f(x) = 1 \) at two points: \( x = 1 \) and \( x \approx 0.3 \).
  **Answer**: \( f(x) = 1 \) at \( x = 0.3 \) and \( x = 1 \).

### (d) Estimate the value of \( x \) such that \( f(x) = 0 \).
- \( f(x) = 0 \) occurs at two points: \( x = 0 \) and \( x \approx 2.1 \).
  **Answer**: \( f(x) = 0 \) at \( x = 0 \) and \( x \approx 2.1 \).

### (e) State the domain and range of \( f \).
- **Domain**: The graph is shown for \( x \) between approximately -1.5 and 2.5. So, the domain is \( [-1.5, 2.5] \).
- **Range**: The values of \( f(x) \) go from 0 to a maximum height of approximately 1.8. So, the range is \( [0, 1.8] \).

### (f) On what interval is \( f \) increasing?
- The function \( f(x) \) is increasing from \( x \approx -1.5 \) to \( x \approx 1.2 \) before it starts decreasing.
  **Answer**: \( f(x) \) is increasing on \( [-1.5, 1.2] \).

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Would you like any additional details or have any questions about this problem? Here are five related questions that could expand on the concepts:

1. How can we formally determine whether a function is increasing or decreasing on an interval?
2. What is the derivative's role in identifying intervals where a function is increasing?
3. How can we refine our estimates for points like \( f(-1) \) and \( f(x) = 0 \) using calculus?
4. What are the maximum and minimum points of the function, and how do they relate to critical points?
5. How would the domain and range change if the graph extended further on either side?

**Tip**: Estimating values from a graph can give approximate solutions, but calculus techniques such as differentiation provide more precise information about increasing/decreasing behavior and critical points.

The graph of a function f is given.
(a) State the value of f(1).
(b) Estimate the value of f(-1).
(c) For what values of x is f(x) = 1?
(d) Estimate the value of x such that f(x) = 0.
(e) State the domain and range of f.
(f) On what interval is f increasing?

Learn how to analyze a function's graph by finding values of f(x), estimating zeros, and identifying the domain, range, and increasing intervals. Includes step-by-step solutions to key problems on function analysis, suitable for high school students.

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