Domain and Range Worksheet #1 | Exercises Calculus | Docsity - Free Printable

Let's analyze Problem #8 from the image you uploaded, as it is clearly visible and has handwritten answers (which we'll verify).

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Graph Description:

The graph shows a horizontal parabola opening to the right, with:
- A solid dot at $ (1, 0) $
- An open circle at $ (5, -2) $
- The curve goes from $ x = 1 $ to $ x = 5 $, but only for certain $ y $-values.

It appears to be a parabola that opens to the right, defined by an equation like $ x = a(y - k)^2 + h $.

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Step-by-step Analysis

#### 1. Domain
The domain is the set of all possible $ x $-values.

From the graph:
- The leftmost point is at $ x = 1 $ (solid dot), so $ x = 1 $ is included.
- The rightmost point is at $ x = 5 $, but it's an open circle, so $ x = 5 $ is not included.
- So, $ x $ ranges from 1 to 5, not including 5.

✔ Domain: $ [1, 5) $

> ✔ Your handwritten answer [1, 5) is correct.

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#### 2. Range
The range is the set of all possible $ y $-values.

Looking at the graph:
- The highest $ y $-value is $ y = 2 $, and it’s reached at $ x = 5 $, but since it's an open circle, $ y = 2 $ is not included.
- The lowest $ y $-value is $ y = -2 $, which is also at $ x = 5 $, open circle → not included.
- But wait — there’s a solid dot at (1, 0), and the graph extends upward and downward from there.

Wait — actually, looking closely:
- The graph starts at $ (1, 0) $ (solid dot).
- Then curves up to $ (5, 2) $ (open circle), and down to $ (5, -2) $ (open circle).
- So, the lowest $ y $-value is $ -2 $, highest is $ 2 $, but both are not included because the endpoints at $ x=5 $ are open circles.
- However, the graph includes $ y = 0 $ at $ x = 1 $, and continues symmetrically.

But here's the key: Is every $ y $-value between $ -2 $ and $ 2 $ covered?

Yes — the graph is a smooth curve from $ (1, 0) $ to $ (5, 2) $ and $ (5, -2) $, forming a sideways parabola.

So:
- $ y $ values go from $ -2 $ to $ 2 $
- But $ y = -2 $ and $ y = 2 $ are not included (since the points at $ x = 5 $ are open)
- $ y = 0 $ is included (at $ x = 1 $)

✔ Range: $ (-2, 2) $

> ✔ Your handwritten answer (-2, 2) is correct.

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#### 3. Is it a Function?

A function must pass the Vertical Line Test: each $ x $-value can have only one $ y $-value.

But in this graph:
- For $ x = 4 $, for example, there are two $ y $-values (one above and one below).
- This means a vertical line would intersect the graph in two places.

✘ Therefore, this is NOT a function.

> ✘ Your handwritten answer says "Function:" — you didn't write anything, but based on the context, you should write:

Function: No

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✔ Final Answers for Problem #8:

- Domain: $ [1, 5) $
- Range: $ (-2, 2) $
- Function: No

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📌 Explanation Summary:

- The graph is a sideways parabola (not a function) because it fails the vertical line test.
- The domain is $ [1, 5) $: includes $ x = 1 $, excludes $ x = 5 $.
- The range is $ (-2, 2) $: $ y $ goes from $ -2 $ to $ 2 $, but neither endpoint is included due to open circles.
- It's not a function because multiple $ y $-values exist for some $ x $-values.

Let me know if you'd like help with other problems on the sheet!