4. $$ p(x)=8 x^{2}-16 x+4 $$ domain: range:

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To determine the **domain** and **range** of the quadratic function $$ p(x) = 8x^{2} - 16x + 4 $$, let's analyze each step by step.

### Domain

The **domain** of a function consists of all possible input values (x-values) for which the function is defined.

For any quadratic function of the form $$ ax^{2} + bx + c $$, the domain is all real numbers because you can substitute any real number for $$ x $$ without causing mathematical issues (like division by zero or taking the square root of a negative number).

**Domain:** 
$$
(-\infty, \infty)
$$

### Range

The **range** of a function consists of all possible output values (y-values) that the function can produce.

1. **Determine the direction of the parabola:**
   - Since the coefficient of $$ x^{2} $$ is positive ($$ a = 8 > 0 $$), the parabola opens **upwards**. This means the function has a **minimum** value.

2. **Find the vertex of the parabola:**
   The vertex of a quadratic function $$ ax^{2} + bx + c $$ occurs at:
   $$
   x = -\frac{b}{2a}
   $$
   Plugging in the values:
   $$
   x = -\frac{-16}{2 \times 8} = \frac{16}{16} = 1
   $$
   
   Now, find the y-coordinate (minimum value) by substituting $$ x = 1 $$ back into $$ p(x) $$:
   $$
   p(1) = 8(1)^{2} - 16(1) + 4 = 8 - 16 + 4 = -4
   $$
   
3. **Determine the range:**
   Since the parabola opens upwards and the minimum value is $$ -4 $$, the range includes all real numbers greater than or equal to $$ -4 $$.

**Range:**
$$
y \geq -4
$$

### Summary

- **Domain:** $$ (-\infty, \infty) $$
- **Range:** $$ y \geq -4 $$
**Domain:** All real numbers **Range:** $$ y \geq -4 $$

4. \( p(x)=8 x^{2}-16 x+4 \) domain: range:

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