Domain and Range of a Function Worksheets - Free Printable

To solve the problem of finding the domain and range for each graph, we need to analyze each graph individually. The domain is the set of all possible \( x \)-values for which the function is defined, and the range is the set of all possible \( y \)-values that the function can take.

Let's go through each graph step by step:

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Graph (a)

#### Description:
- The graph is a straight line segment with endpoints at \((-3, 2)\) and \((4, -1)\).

#### Domain:
- The \( x \)-values range from \(-3\) to \(4\), inclusive.
- Therefore, the domain is: \([-3, 4]\).

#### Range:
- The \( y \)-values range from \(-1\) to \(2\), inclusive.
- Therefore, the range is: \([-1, 2]\).

#### Final Answer for Graph (a):
- Domain: \([-3, 4]\)
- Range: \([-1, 2]\)

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Graph (b)

#### Description:
- The graph is a parabola opening upwards with its vertex at \((0, -2)\). The parabola extends infinitely in both directions along the \( x \)-axis.

#### Domain:
- Since the parabola extends infinitely to the left and right, the \( x \)-values can be any real number.
- Therefore, the domain is: \((-\infty, \infty)\).

#### Range:
- The vertex of the parabola is at \((0, -2)\), and since the parabola opens upwards, the minimum \( y \)-value is \(-2\). The \( y \)-values increase without bound as \( x \) moves away from the vertex.
- Therefore, the range is: \([-2, \infty)\).

#### Final Answer for Graph (b):
- Domain: \((-\infty, \infty)\)
- Range: \([-2, \infty)\)

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Graph (c)

#### Description:
- The graph is a horizontal line at \( y = 3 \). This line extends infinitely in both directions along the \( x \)-axis.

#### Domain:
- Since the line extends infinitely to the left and right, the \( x \)-values can be any real number.
- Therefore, the domain is: \((-\infty, \infty)\).

#### Range:
- The line is horizontal at \( y = 3 \), so the \( y \)-value is always \(3\).
- Therefore, the range is: \(\{3\}\).

#### Final Answer for Graph (c):
- Domain: \((-\infty, \infty)\)
- Range: \(\{3\}\)

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Graph (d)

#### Description:
- The graph is a parabola opening downwards with its vertex at \((2, 5)\). The parabola extends infinitely in both directions along the \( x \)-axis.

#### Domain:
- Since the parabola extends infinitely to the left and right, the \( x \)-values can be any real number.
- Therefore, the domain is: \((-\infty, \infty)\).

#### Range:
- The vertex of the parabola is at \((2, 5)\), and since the parabola opens downwards, the maximum \( y \)-value is \(5\). The \( y \)-values decrease without bound as \( x \) moves away from the vertex.
- Therefore, the range is: \((-\infty, 5]\).

#### Final Answer for Graph (d):
- Domain: \((-\infty, \infty)\)
- Range: \((-\infty, 5]\)

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Summary of Answers

1. Graph (a):
- Domain: \([-3, 4]\)
- Range: \([-1, 2]\)

2. Graph (b):
- Domain: \((-\infty, \infty)\)
- Range: \([-2, \infty)\)

3. Graph (c):
- Domain: \((-\infty, \infty)\)
- Range: \(\{3\}\)

4. Graph (d):
- Domain: \((-\infty, \infty)\)
- Range: \((-\infty, 5]\)

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Final Boxed Answers:

\[
\boxed{
\begin{array}{ll}
\text{Graph (a)} & \text{Domain: } [-3, 4], \text{ Range: } [-1, 2] \\
\text{Graph (b)} & \text{Domain: } (-\infty, \infty), \text{ Range: } [-2, \infty) \\
\text{Graph (c)} & \text{Domain: } (-\infty, \infty), \text{ Range: } \{3\} \\
\text{Graph (d)} & \text{Domain: } (-\infty, \infty), \text{ Range: } (-\infty, 5] \\
\end{array}
}
\]