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Before students learn about functions, they typically learn about relating two variables. This is most obvious in the context of slope-intercept form of a line, which relates y to x, using the slope and the y-intercept.

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Once again, slope-intercept form of a line is written as y = mx+b.

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Another way that we could write this same equation is using function notation. This would look like f(x) = mx+b.

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Notice that f(x) and y are the same thing!

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For the function f(x) , there are two distinguishing features:

- The argument or input
- The function name or the output

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The argument is the value that you put into the function. For f(x) , the argument is just x.

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For this function, the function name is simply f and the output of the function is f(x).

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A common misconception that students sometimes have is that function notation means to multiply f by x. This is because the parentheses typically mean multiplication in other contexts. However, in this context, we must be sure to understand that the notation f(x) means to substitute a value for x into the function f.

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Functions can also have any name that you want, but are most often written using f(x), g(x) or h(x).

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An interesting application for functions is to describe the areas of specific polygons.

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For instance, a square would have an area function that looks like: *A*(* _{s}*) =

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We could understand that the area function A takes in the side length of our square, which is s, and then squares it. This means that the area of a square is a function of its side length!

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Using this for a specific side length of a square gives us a numeric value for the function. If a square has a side length of 6 inches, we could use the function to find the specific area.

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Substitute the value 6 for s to find that A(6) = 6^{2} = 36. Therefore, our square has an area of 36 square inches.

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For more abstract functions, we may not be able to fully describe what our function represents. For instance, the function is not an obvious representation of area, perimeter, volume, or anything else that immediately comes to mind.

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However, we can still substitute values for x into this function to find the value of f at those values.

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When substituting values for x, be sure to use parentheses for each instance of x as this will prevent you from making any serious errors.

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For the function , we can substitute x = 3 to find the following:

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Notice that for every instance of 3, we used parentheses. This is very important when you begin to plug in negative numbers for functions!

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If we had chosen to plug in x = -2 instead, the parentheses are even more useful.

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This would show us that:

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For any function, you can always just substitute the value of the argument, which is usually x, to find the value of the function at that point.

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Always think of functions in terms of inputting a value x and receiving a value of the function f(x) as an output and you’ll be in good shape!

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