We are tasked with graphing the function $g(x) = x^2 - 2$, and analyzing its domain and range using transformations. Let's break this process down into steps, starting with the basic function.

1. Start with the basic function:

The basic function here is $f(x) = x^2$, which is a parabola that opens upwards with its vertex at the origin $(0,0)$.

2. Identify transformations:

The function $g(x) = x^2 - 2$ is a transformation of the basic function $f(x) = x^2$. Specifically:

• Vertical Shift: The term $-2$ represents a vertical shift down by 2 units.

Thus, the function $g(x) = x^2 - 2$ will be the same parabola as $f(x) = x^2$, but it will be shifted downwards by 2 units. The vertex of the new parabola will be at $(0, -2)$.

3. Graphing the function:

• Start with the parabola $f(x) = x^2$, which has a vertex at $(0, 0)$ and passes through points like $(-1, 1)$, $(1, 1)$, $(-2, 4)$, and $(2, 4)$.
• Now shift the entire graph down by 2 units. This moves the vertex to $(0, -2)$, and the graph passes through points $(-1, -1)$, $(1, -1)$, $(-2, 2)$, and $(2, 2)$.

4. Domain and Range:

• Domain: The domain of $g(x) = x^2 - 2$ is the same as the domain of $f(x) = x^2$, which is all real numbers. Therefore, the domain is $(-\infty, \infty)$.
• Range: Since the parabola opens upwards and the lowest point is the vertex $(0, -2)$, the range is $[-2, \infty)$.

Summary:

• The function $g(x) = x^2 - 2$ is a downward vertical shift of the basic parabola $f(x) = x^2$.
• The domain is $(-\infty, \infty)$, and the range is $[-2, \infty)$.

Do you have any questions or would you like more details on any of these steps?

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Here are 5 related questions to deepen understanding:

1. How would the graph change if we had $g(x) = x^2 + 3$?
2. What would be the domain and range of $g(x) = -x^2 - 2$?
3. How does the graph change when we add or subtract a constant to a function?
4. How can we reflect the graph of $x^2$ over the x-axis?
5. What happens to the graph of $g(x) = x^2$ if we compress it horizontally?

Tip: Always check how constants outside or inside the function (like $+c$ or $-c$) affect the graph through transformations!