What similarities and differences do you see between functions and linear equations studied in Ch. 3? Are all linear equations functions? Is there an instance when a linear equation is not a function? Support your answer. Create an equation of a nonlinear function and provide two inputs for your classmates to evaluate.

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To compare the similarities and differences between functions and linear equations studied in Chapter 3, let's start by understanding what functions and linear equations are.

A function is a relation between a set of inputs (known as the domain) and a set of outputs (known as the range), where each input value relates to exactly one output value. In other words, for each input, there is only one corresponding output.

On the other hand, a linear equation is an equation that represents a straight line on a graph. It is written in the form y = mx + b, where m represents the slope of the line, and b represents the y-intercept (the point where the line crosses the y-axis).

Now, let's address the first part of your question: Are all linear equations functions? The answer is yes. All linear equations are functions because, for any given x-value, there is only one corresponding y-value. In other words, there are no horizontal lines in a linear equation, so every x-value has a unique y-value.

However, there can be instances when a linear equation is not a function. This occurs when a vertical line is formed on the graph. For example, the equation x = 2 represents a vertical line where every y-value corresponds to x = 2. Since there are infinitely many y-values for a single x-value, this violates the definition of a function.

Now, let's create an equation of a nonlinear function and provide two inputs for your classmates to evaluate:

Nonlinear function: y = x^2

Input 1: x = 3
To evaluate this, substitute x = 3 into the equation:
y = (3)^2
y = 9

Input 2: x = -2
To evaluate this, substitute x = -2 into the equation:
y = (-2)^2
y = 4

So, for the nonlinear function y = x^2, when x = 3, y = 9, and when x = -2, y = 4.