The problem asks us to determine whether the given functions are linear or nonlinear. The functions are: $y = \frac{2}{x} - 3$ $y = \frac{1}{3}x + 3$ $y = 4x$ $y = x^2 + 4$

1. Problem Description

The problem asks us to determine whether the given functions are linear or nonlinear. The functions are:

$y = \frac{2}{x} - 3$

$y = \frac{1}{3}x + 3$

$y = 4x$

$y = x^2 + 4$

2. Solution Steps

A linear function has the general form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The exponent of $x$ must be

1. - Function 1: $y = \frac{2}{x} - 3 = 2x^{-1} - 3$. The exponent of $x$ is -1, so this is a nonlinear function.

- Function 2: $y = \frac{1}{3}x + 3$. This is in the form $y = mx + b$ with $m = \frac{1}{3}$ and $b = 3$. Thus, this is a linear function.

- Function 3: $y = 4x$. This is in the form $y = mx + b$ with $m = 4$ and $b = 0$. Thus, this is a linear function.

- Function 4: $y = x^2 + 4$. The exponent of $x$ is 2, so this is a nonlinear function.

3. Final Answer

- $y = \frac{2}{x} - 3$: Nonlinear

- $y = \frac{1}{3}x + 3$: Linear

- $y = 4x$: Linear

- $y = x^2 + 4$: Nonlinear