Problem Analysis:
The task involves analyzing a given graph of a function \( f \) to determine specific values of the function and its domain. Let's solve each part step by step.
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Graph Description:
- The graph shows a piecewise function.
- There is a straight line segment extending from the bottom left to the point \((-1, 0)\).
- At \( x = -1 \), there is an open circle, indicating that the function value at \( x = -1 \) is not included.
- From \( x = -1 \) to \( x = 2 \), the graph is a horizontal line at \( y = 0 \).
- At \( x = 2 \), there is a closed circle, indicating that the function value at \( x = 2 \) is included.
- From \( x = 2 \) onward, the graph is a downward-opening parabola.
Now, let's solve each part of the problem.
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(a) Determine \( f(-2) \)
1. Locate \( x = -2 \) on the graph.
2. Follow the vertical line at \( x = -2 \) until it intersects the graph.
3. The graph shows that at \( x = -2 \), the function value is on the straight line segment.
4. The corresponding \( y \)-value is \( 0 \).
Thus, \( f(-2) = 0 \).
Answer: \( \boxed{0} \)
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(b) Determine \( f(0.2) \)
1. Locate \( x = 0.2 \) on the graph.
2. Follow the vertical line at \( x = 0.2 \) until it intersects the graph.
3. The graph shows that at \( x = 0.2 \), the function value is on the horizontal line segment between \( x = -1 \) and \( x = 2 \).
4. The corresponding \( y \)-value is \( 0 \).
Thus, \( f(0.2) = 0 \).
Answer: \( \boxed{0} \)
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(c) Determine \( f(3) \)
1. Locate \( x = 3 \) on the graph.
2. Follow the vertical line at \( x = 3 \) until it intersects the graph.
3. The graph shows that at \( x = 3 \), the function value is on the downward-opening parabola.
4. By observation, the \( y \)-value at \( x = 3 \) appears to be approximately \(-1\).
Thus, \( f(3) = -1 \).
Answer: \( \boxed{-1} \)
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(d) Determine the domain
1. The domain of a function is the set of all possible input values (\( x \)-values) for which the function is defined.
2. From the graph:
- The function is defined for all \( x \)-values starting from negative infinity (\( -\infty \)).
- There are no breaks or gaps in the graph along the \( x \)-axis.
- The graph extends indefinitely to the right.
Thus, the domain is all real numbers, which can be written as \( (-\infty, \infty) \).
Answer: \( \boxed{(-\infty, \infty)} \)
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Final Answers:
(a) \( \boxed{0} \)
(b) \( \boxed{0} \)
(c) \( \boxed{-1} \)
(d) \( \boxed{(-\infty, \infty)} \)
Parent Tip: Review the logic above to help your child master the concept of finding domain and range from a graph worksheet.
