The problem requires us to analyze the transformation of a parabola from its parent function $f(x) = x^2$ to a new function $g(x)$. We are given that there's no vertical stretch or compression, a vertical shift down by 4 units, and a horizontal shift to the right by 5 units. We need to express $g(x)$ in function notation and as an equation, and then determine its domain and range using interval notation.

1. Problem Description

The problem requires us to analyze the transformation of a parabola from its parent function $f(x) = x^2$ to a new function $g(x)$. We are given that there's no vertical stretch or compression, a vertical shift down by 4 units, and a horizontal shift to the right by 5 units. We need to express $g(x)$ in function notation and as an equation, and then determine its domain and range using interval notation.

2. Solution Steps

First, let's consider the parent function $f(x) = x^2$.

We are told the graph has been shifted horizontally right by 5 units. This can be expressed as $f(x-5) = (x-5)^2$.

We are also told the graph has been shifted vertically down by 4 units. This can be expressed as $f(x-5) - 4 = (x-5)^2 - 4$.

Therefore, we can write $g(x) = f(x-5) - 4 = (x-5)^2 - 4$.

Now, let's consider the domain of $g(x)$. Since $g(x)$ is a quadratic function, it is defined for all real numbers. Thus, the domain is $(-\infty, \infty)$.

Next, let's consider the range of $g(x)$. The vertex of the parabola $g(x) = (x-5)^2 - 4$ is at $(5, -4)$. Since the coefficient of the $x^2$ term is positive, the parabola opens upwards. Therefore, the minimum value of $g(x)$ is $-4$, and it extends to infinity. Thus, the range is $[-4, \infty)$.

4) Function notation: $g(x)$

Since $g(x)$ is a transformation of $f(x)=x^2$ with a horizontal shift right 5 and a vertical shift down 4, $g(x)=f(x-5)-4$.

5) As an equation: $g(x)$

From the transformations, $g(x) = (x-5)^2 - 4$.

6) Domain of $g(x)$:

Since the quadratic is defined for all real numbers, the domain is $(-\infty, \infty)$.

7) Range of $g(x)$:

The minimum value of $g(x)$ is -4 (at the vertex), and the values go up to infinity. Thus, the range is $[-4, \infty)$.

3. Final Answer

$g(x) = f(x-5) - 4$

$g(x) = (x-5)^2 - 4$

Domain of $g(x)$: $(-\infty, \infty)$

Range of $g(x)$: $[-4, \infty)$