Question

What are the differences among expressions, equations, and functions? Provide examples of each

Answer 1

This is an excellent question. It is OFTEN overlooked that an equation will somehow involve this symbol, "=".

Answer 2

(\( \textbf{Warning: }\)This post is very long! My goal is to try and explain those terms as simply as possible, so try not to be intimidated by the length! Thanks! :) )
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If you want to compare the words, it helps to know what each of the words mean!
A lot of times, we can identify these mathematical words a lot like the same words in a regular, everyday sort of context.

Expression:
- In the general sense, an expression is just a phrase or sort of statement. It's a single 'thing.' For example, "I like your necklace" is a simple expression -- it says one tihng.
- Mathematically, we refer to expressions as single mathematical objects. We talk about a distinct object like "1," "5x + 7," or even something more complicated like "sin(2x) + 6x - 1/x" all as single items. We don't say anything about what they really do.


Equation:
- We can extract that "equa" piece from the word. It sounds a lot like "equal." Unsurprisingly, this is all an equation does. It's showing that one thing -- one expression -- is equal to another.
- We could say: 1 + 1 = 2. This is an equation that says, 1 + 1 is the same as 2.
- We could also say: 6x = 5x + 1. This is another equation with different meaning, that says tat 6x is the same as 5x + 1.
--- Notice that it makes sense that, if they have exactly the same value on each side, then it won't change by changing both sides equivalently.


Function:
- In a general sense, function just menas some object does something, right?
- In the mathematical sense, a function is a bit more specific. If we feed it a value, it 'functions' by applying certain rules to that input to give an output.
f(x) = 5x + 2. If we give it x=2, then it functions by taking in x=2, and spitting out f(2) = 5*2 + 2 = 12.

Answer 3

Interestingly, we can see that expressions are sort of the building block for the two other topics.
An equation is created by two expressions and an equals-sign.
A function is a special form of equation with its simple expression f(x) (function notation) being one thing and the other expression defining what that function means.

You may be more familiar with using y instead of f(x), which is fine as well.