Domain And Range Practice Worksheet Elegant Honors Precalc ... - 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 produce.

Let's go through each graph step by step:

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Graph 1:

- Shape: A parabola opening upwards.
- Domain: The graph extends infinitely to the left and right along the \( x \)-axis. Therefore, the domain is all real numbers.
\[
\text{Domain: } (-\infty, \infty)
\]
- Range: The vertex of the parabola is at the lowest point, which appears to be at \( y = -3 \). The graph extends upwards infinitely. Therefore, the range is all \( y \)-values greater than or equal to \(-3\).
\[
\text{Range: } [-3, \infty)
\]

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Graph 2:

- Shape: A hyperbola with two branches.
- Domain: The graph has vertical asymptotes at \( x = -2 \) and \( x = 2 \). The function is defined for all \( x \)-values except \( x = -2 \) and \( x = 2 \).
\[
\text{Domain: } (-\infty, -2) \cup (-2, 2) \cup (2, \infty)
\]
- Range: The graph has horizontal asymptotes at \( y = 0 \). The function approaches \( y = 0 \) but never reaches it. The branches extend infinitely upwards and downwards. Therefore, the range is all real numbers except \( y = 0 \).
\[
\text{Range: } (-\infty, 0) \cup (0, \infty)
\]

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Graph 3:

- Shape: A parabola opening downwards.
- Domain: The graph extends infinitely to the left and right along the \( x \)-axis. Therefore, the domain is all real numbers.
\[
\text{Domain: } (-\infty, \infty)
\]
- Range: The vertex of the parabola is at the highest point, which appears to be at \( y = 4 \). The graph extends downwards infinitely. Therefore, the range is all \( y \)-values less than or equal to \( 4 \).
\[
\text{Range: } (-\infty, 4]
\]

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Graph 4:

- Shape: A straight line.
- Domain: The graph extends infinitely to the left and right along the \( x \)-axis. Therefore, the domain is all real numbers.
\[
\text{Domain: } (-\infty, \infty)
\]
- Range: The graph extends infinitely upwards and downwards along the \( y \)-axis. Therefore, the range is all real numbers.
\[
\text{Range: } (-\infty, \infty)
\]

---

Graph 5:

- Shape: A semicircle.
- Domain: The semicircle spans from \( x = -3 \) to \( x = 3 \). Therefore, the domain is:
\[
\text{Domain: } [-3, 3]
\]
- Range: The semicircle spans from \( y = -3 \) to \( y = 0 \). Therefore, the range is:
\[
\text{Range: } [-3, 0]
\]

---

Graph 6:

- Shape: A V-shaped graph (absolute value function).
- Domain: The graph extends infinitely to the left and right along the \( x \)-axis. Therefore, the domain is all real numbers.
\[
\text{Domain: } (-\infty, \infty)
\]
- Range: The vertex of the V-shape is at \( y = 0 \), and the graph extends upwards infinitely. Therefore, the range is all \( y \)-values greater than or equal to \( 0 \).
\[
\text{Range: } [0, \infty)
\]

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Final Answers:

1. Domain: \( (-\infty, \infty) \), Range: \( [-3, \infty) \)
2. Domain: \( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \), Range: \( (-\infty, 0) \cup (0, \infty) \)
3. Domain: \( (-\infty, \infty) \), Range: \( (-\infty, 4] \)
4. Domain: \( (-\infty, \infty) \), Range: \( (-\infty, \infty) \)
5. Domain: \( [-3, 3] \), Range: \( [-3, 0] \)
6. Domain: \( (-\infty, \infty) \), Range: \( [0, \infty) \)

\[
\boxed{
\begin{array}{ll}
1. & \text{Domain: } (-\infty, \infty), \text{Range: } [-3, \infty) \\
2. & \text{Domain: } (-\infty, -2) \cup (-2, 2) \cup (2, \infty), \text{Range: } (-\infty, 0) \cup (0, \infty) \\
3. & \text{Domain: } (-\infty, \infty), \text{Range: } (-\infty, 4] \\
4. & \text{Domain: } (-\infty, \infty), \text{Range: } (-\infty, \infty) \\
5. & \text{Domain: } [-3, 3], \text{Range: } [-3, 0] \\
6. & \text{Domain: } (-\infty, \infty), \text{Range: } [0, \infty) \\
\end{array}
}
\]