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).