The problem requires us to express the transformed function $g(x)$ in both function notation and equation form, given the parent function $f(x)$ and a series of transformations. We are given three different parent functions and their corresponding transformations.

1. Problem Description

The problem requires us to express the transformed function $g(x)$ in both function notation and equation form, given the parent function $f(x)$ and a series of transformations. We are given three different parent functions and their corresponding transformations.

2. Solution Steps

Case 1: $f(x) = x^3$

Vertical shift up by 3: $f(x) + 3$

Horizontal shift left by 4: $f(x + 4)$

Combining the transformations: $f(x + 4) + 3$

Function Notation: $g(x) = f(x + 4) + 3$

Equation Form: $g(x) = (x + 4)^3 + 3$

Case 2: $f(x) = x^2$

Vertical stretch by a factor of 3: $3f(x)$

Vertical shift down by 2: $3f(x) - 2$

Horizontal shift left by 1: $3f(x + 1) - 2$

Function Notation: $g(x) = 3f(x + 1) - 2$

Equation Form: $g(x) = 3(x + 1)^2 - 2$

Case 3: $f(x) = \sqrt{x}$

Vertical stretch by a factor of $\frac{4}{3}$: $\frac{4}{3}f(x)$

Vertical shift down by 1: $\frac{4}{3}f(x) - 1$

Horizontal shift right by 2: $\frac{4}{3}f(x - 2) - 1$

Function Notation: $g(x) = \frac{4}{3}f(x - 2) - 1$

Equation Form: $g(x) = \frac{4}{3}\sqrt{x - 2} - 1$

3. Final Answer

Case 1:

Function Notation: $g(x) = f(x + 4) + 3$

Equation Form: $g(x) = (x + 4)^3 + 3$

Case 2:

Function Notation: $g(x) = 3f(x + 1) - 2$

Equation Form: $g(x) = 3(x + 1)^2 - 2$

Case 3:

Function Notation: $g(x) = \frac{4}{3}f(x - 2) - 1$

Equation Form: $g(x) = \frac{4}{3}\sqrt{x - 2} - 1$