https://file.aiquickdraw.com/mathbot/file/1727336427002bdsv4z7o.jpg

Let's work on the two parts of the problem:

### (a) \( g(x) = 4x^2 - 8 \)

#### Domain:
The function \( g(x) = 4x^2 - 8 \) is a quadratic function, and quadratic functions are defined for all real numbers. Hence, the **domain** is:
\[
\text{Domain of } g(x) = (-\infty, \infty)
\]

#### Range:
The function \( g(x) = 4x^2 - 8 \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive). The minimum value of this function occurs at \( x = 0 \), which is the vertex. Plugging \( x = 0 \) into the function:
\[
g(0) = 4(0)^2 - 8 = -8
\]
Thus, the minimum value of \( g(x) \) is \( -8 \), and since the parabola opens upwards, the function takes values greater than or equal to \( -8 \). Therefore, the **range** is:
\[
\text{Range of } g(x) = [-8, \infty)
\]

---

### (b) \( h(x) = \{ (-6, 2), (-9, 0), (-2, -1), (3, 4) \} \)

Here, \( h(x) \) is given as a set of ordered pairs.

#### Domain:
The domain consists of all the \( x \)-values from the ordered pairs. Extracting these values:
\[
\text{Domain of } h(x) = \{ -6, -9, -2, 3 \}
\]

#### Range:
The range consists of all the \( y \)-values from the ordered pairs. Extracting these values:
\[
\text{Range of } h(x) = \{ 2, 0, -1, 4 \}
\]

---

### Summary:
- **(a)**
  - Domain of \( g(x) \): \( (-\infty, \infty) \)
  - Range of \( g(x) \): \( [-8, \infty) \)

- **(b)**
  - Domain of \( h(x) \): \( \{ -6, -9, -2, 3 \} \)
  - Range of \( h(x) \): \( \{ 2, 0, -1, 4 \} \)

Would you like further details or have any questions?

---

Here are 5 related questions you could explore:
1. How do you find the domain and range of more complex functions like rational or logarithmic ones?
2. What is the significance of the vertex in the range of quadratic functions?
3. Can you explain how to determine whether a function is continuous over its domain?
4. How does the graph of \( g(x) = 4x^2 - 8 \) compare to the graph of \( f(x) = x^2 \)?
5. How would you find the inverse of a function if it exists?

**Tip:** For polynomial functions like quadratics, the domain is usually all real numbers unless specified otherwise by restrictions (e.g., division by zero or square roots).

Find the domain and range of each function (a) g(x) = 4x^2 - 8 and (b) h(x) = {(-6, 2), (-9, 0), (-2, -1), (3, 4)}

Learn how to find the domain and range of a quadratic function, g(x) = 4x^2 - 8, and a set of ordered pairs, h(x). Explore key mathematical concepts such as quadratic functions, domains, and ranges. This problem is ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation