Worksheet for graphing radical functions with domain and range analysis.

Let's solve each of these radical function problems step by step. We'll:

1. Graph the function (described verbally since we can't draw here),
2. Determine the domain and range in interval notation,
3. Explain the reasoning.

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🔷 General Rules for Radical Functions:

- For square roots $ \sqrt{x} $:
- Domain: $ x \geq 0 $ → because you can't take the square root of a negative number in real numbers.
- Range depends on transformations.

- For cube roots $ \sqrt[3]{x} $:
- Domain: All real numbers $ (-\infty, \infty) $
- Range: All real numbers

- Transformations:
- $ f(x) = \sqrt{x - h} + k $: Shift right by $ h $, up by $ k $
- $ f(x) = a\sqrt{x} $: Vertical stretch/shrink; if $ a < 0 $, reflection over x-axis

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Now let’s go through each one.

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① $ f(x) = -\sqrt{x} $

#### ✔ Graphing:
- Start with $ y = \sqrt{x} $: starts at (0,0), increases slowly to the right.
- Multiply by -1 → reflect over the x-axis → opens downward.

So this is the negative square root of $ x $, starting at (0,0), going down to the right.

#### ✔ Domain:
We need $ x \geq 0 $ → domain: $ [0, \infty) $

#### ✔ Range:
Since $ \sqrt{x} \geq 0 $, then $ -\sqrt{x} \leq 0 $ → range: $ (-\infty, 0] $

> ✔️ Domain: $ [0, \infty) $
> ✔️ Range: $ (-\infty, 0] $

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② $ f(x) = \sqrt{x} - 2 $

#### ✔ Graphing:
- Start with $ y = \sqrt{x} $
- Subtract 2 → shift down by 2 units
- So it starts at (0, -2)

#### ✔ Domain:
Still $ x \geq 0 $ → $ [0, \infty) $

#### ✔ Range:
$ \sqrt{x} \geq 0 $ → $ \sqrt{x} - 2 \geq -2 $ → range: $ [-2, \infty) $

> ✔️ Domain: $ [0, \infty) $
> ✔️ Range: $ [-2, \infty) $

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③ $ f(x) = \sqrt{x + 1} - 2 $

#### ✔ Graphing:
- Start with $ y = \sqrt{x} $
- Replace $ x $ with $ x+1 $ → shift left by 1 unit
- Then subtract 2 → shift down by 2 units
- Vertex at $ (-1, -2) $

#### ✔ Domain:
Inside radical: $ x + 1 \geq 0 $ → $ x \geq -1 $ → domain: $ [-1, \infty) $

#### ✔ Range:
$ \sqrt{x+1} \geq 0 $ → $ \sqrt{x+1} - 2 \geq -2 $ → range: $ [-2, \infty) $

> ✔️ Domain: $ [-1, \infty) $
> ✔️ Range: $ [-2, \infty) $

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④ $ f(x) = -2 + \sqrt[3]{x} $

This is a cube root function.

#### ✔ Graphing:
- Basic cube root: $ y = \sqrt[3]{x} $, passes through origin, increasing, odd symmetry.
- Add $ -2 $ → shift down by 2 units.
- So now passes through $ (0, -2) $

Cube roots are defined for all real numbers.

#### ✔ Domain:
All real numbers → $ (-\infty, \infty) $

#### ✔ Range:
Cube roots output all real numbers → range: $ (-\infty, \infty) $

> ✔️ Domain: $ (-\infty, \infty) $
> ✔️ Range: $ (-\infty, \infty) $

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⑤ $ f(x) = -3\sqrt[3]{x} $

Cube root again.

#### ✔ Graphing:
- $ y = \sqrt[3]{x} $ → basic shape
- Multiply by $ -3 $:
- Reflect over x-axis (because of negative)
- Stretch vertically by factor of 3

So steeper and flipped.

#### ✔ Domain:
All real numbers → $ (-\infty, \infty) $

#### ✔ Range:
Still all real numbers → $ (-\infty, \infty) $

> ✔️ Domain: $ (-\infty, \infty) $
> ✔️ Range: $ (-\infty, \infty) $

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⑥ $ f(x) = \sqrt{3x - 6} $

Square root — need expression inside ≥ 0.

#### ✔ Graphing:
- Simplify: $ \sqrt{3(x - 2)} $
- So it's a horizontal compression and shift
- Let’s find where it starts: $ 3x - 6 \geq 0 $ → $ x \geq 2 $
- At $ x = 2 $, $ f(2) = \sqrt{0} = 0 $
- As $ x $ increases, function increases

It's a square root function shifted and stretched.

#### ✔ Domain:
$ 3x - 6 \geq 0 $ → $ x \geq 2 $ → $ [2, \infty) $

#### ✔ Range:
$ \sqrt{3x - 6} \geq 0 $ → $ [0, \infty) $

> ✔️ Domain: $ [2, \infty) $
> ✔️ Range: $ [0, \infty) $

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✔ Final Answers Summary:

| Problem | Function | Domain | Range |
|--------|---------|--------|-------|
| ① | $ f(x) = -\sqrt{x} $ | $ [0, \infty) $ | $ (-\infty, 0] $ |
| ② | $ f(x) = \sqrt{x} - 2 $ | $ [0, \infty) $ | $ [-2, \infty) $ |
| ③ | $ f(x) = \sqrt{x + 1} - 2 $ | $ [-1, \infty) $ | $ [-2, \infty) $ |
| ④ | $ f(x) = -2 + \sqrt[3]{x} $ | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ |
| ⑤ | $ f(x) = -3\sqrt[3]{x} $ | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ |
| ⑥ | $ f(x) = \sqrt{3x - 6} $ | $ [2, \infty) $ | $ [0, \infty) $ |

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📌 How to Graph (Tips):

For each:
- Find the starting point (where the expression under the radical is zero).
- Plot a few points.
- Use transformations:
- $ \sqrt{x} $ → shift left/right, up/down, stretch/reflect.
- $ \sqrt[3]{x} $ → same but no restrictions.

If you’re graphing:
- For square roots: start at domain endpoint, curve upward or downward.
- For cube roots: pass through the transformed origin, symmetric.

Let me know if you want a visual sketch description for any!