Getting the Domain and Range from the Graph of a Piecewise ... - Free Printable

Looking at the graph, we can see it’s a piecewise function — meaning it’s made of different parts (or pieces) defined over different intervals.

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Step 1: Identify the pieces

There are two distinct parts to this graph:

1. A straight line segment (red) on the left.
2. A curved parabola-like segment (black) on the right.

We also notice open circles at the endpoints of each piece — this means those points are not included in that piece.

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Step 2: Determine the domain for each piece

- The red line goes from `x = -8` to `x = -4`.
- At `x = -8`, there’s an open circle → so `x = -8` is not included.
- At `x = -4`, there’s an open circle → so `x = -4` is not included.
- So the red line is defined for:
`-8 < x < -4`

- The black curve starts at `x = -4` and goes to `x = 4`.
- At `x = -4`, open circle → not included.
- At `x = 4`, open circle → not included.
- So the black curve is defined for:
`-4 < x < 4`

> ⚠️ Note: There’s no point defined at `x = -4` — both pieces have open circles there, so the function is undefined at x = -4.

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Step 3: Write the piecewise function

We need to find equations for each piece.

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#### ➤ Red Line Segment: `-8 < x < -4`

Points on the line:
- Open circle at `(-8, -6)` — not included
- Open circle at `(-4, -2)` — not included

Even though they’re not included, we can use them to find the slope.

Slope `m = (y₂ - y₁)/(x₂ - x₁) = (-2 - (-6)) / (-4 - (-8)) = (4)/(4) = 1`

So slope is 1.

Use point-slope form with point `(-4, -2)`:

`y - (-2) = 1(x - (-4))`
→ `y + 2 = x + 4`
→ `y = x + 2`

✔ So for `-8 < x < -4`, the function is:
`f(x) = x + 2`

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#### ➤ Black Curve: `-4 < x < 4`

This looks like a parabola opening upwards, vertex at `(0, 0)`.

General form: `f(x) = ax²`

We can use a point on the curve to find `a`.

Look at point `(4, 16)` — but it’s an open circle, so not included, but still lies on the curve.

Plug into `f(x) = ax²`:

`16 = a*(4)² = 16a` → `a = 1`

✔ So for `-4 < x < 4`, the function is:
`f(x) = x²`

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✔ Final Answer: Piecewise Function

```
{ x + 2, if -8 < x < -4
f(x) = {
{ x², if -4 < x < 4
```

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📌 Additional Notes:

- The function is undefined at x = -4, since both pieces have open circles there.
- Domain of the entire function: `(-8, -4) ∪ (-4, 4)`
- Range: For the red line (`x+2`), as `x` goes from -8 to -4, `y` goes from -6 to -2 → range `(-6, -2)`
For the black curve (`x²`), as `x` goes from -4 to 4, `y` goes from 0 to 16 → range `[0, 16)` (since 0 is achieved at x=0, but 16 is not included)
So overall range: `(-6, -2) ∪ [0, 16)`

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Let me know if you’d like to evaluate the function at specific points or analyze continuity/differentiability!