What similarities and differences do you see between functions and linear equations, Are all linear equations functions? Is there an instance when a linear equation is not a function? Create an equation of a nonlinear function and provide two inputs to evaluate.

A straight line, unless vertical, has one and only one value of the dependent variable (y for example) for every value of the independent variable (x for example). It is therefore a function unless it is vertical. Therefore I claim that all linear equations are functions except one of form x = constant.

An easy non-linear function is y = x^2
Here there is one and only one value of y for every value of x so it is a function. You can put in a couple of values of x and find y, for example (0,0) and (3,9).
Notice that x is NOT a function of y in this equation because there are TWO values of x for every y.

Similarities between functions and linear equations:

- Both functions and linear equations represent relationships between variables.
- They can both be used to solve problems and make predictions.
- Both functions and linear equations can be represented graphically.

Differences between functions and linear equations:
- A function is a relation where each input (or x-value) is associated with exactly one output (or y-value). On the other hand, a linear equation represents a straight line on a graph.
- Functions can be linear or nonlinear, but linear equations are specifically equations that represent a straight line.
- Functions can have various forms, such as polynomial, exponential, or logarithmic, while linear equations have a specific form of y = mx + b, where m is the slope and b is the y-intercept.

Not all linear equations are functions. A linear equation is not a function if it fails the vertical line test. The vertical line test states that if a vertical line intersects a graph at more than one point, then the relation (or equation) is not a function. In other words, if there are two or more x-values that correspond to the same y-value, the graph fails the vertical line test and is not a function. This can happen in cases where a vertical line intersects a straight line more than once.

Here's an example of a nonlinear function:
y = x^2 - 3x + 2

To evaluate this function, we can substitute two different inputs (x-values) and find the corresponding outputs (y-values). Let's use x = 1 and x = 3.

For x = 1:
y = 1^2 - 3(1) + 2
= 1 - 3 + 2
= 0

So, when x = 1, y = 0.

For x = 3:
y = 3^2 - 3(3) + 2
= 9 - 9 + 2
= 2

Therefore, when x = 3, y = 2.

So, for the given nonlinear function y = x^2 - 3x + 2, when x = 1, y = 0; and when x = 3, y = 2.