<< Back to Blog#### An Introduction to Functions

###### Sep-22-2021 | Hits: 282 |

##### Linear Polynomial Functions

##### Quadratic Polynomial Functions

##### Cubic Polynomial Functions

A function describes a relationship between two or more variables. Consider the equation `y=x+1`. This equation describes a relationship between the variable `y` and the variable `x`, and it tells you that the value of `y` is always `1` more than the value of `x`.

The syntax `f(x)` tells you that `f` is a function, and it takes `x` as an input. `f(x)` defines a relationship between two things that are represented mathematically by variables, `x` and `y`, where `y=f(x)`. I have seen many students confused by the function notation, `f(x)`. I think it's just a result of not having much experience with functions, or maybe having a previous teacher who wasn't good at explaining what they are, or both. Anyway, when you see `f(x)` it is pronounced, "`f` of `x`". The `f` and the `x` are not separate; it's not `f` times `x` as some students want to think. They are one thing that defines a relationship between a variable `x` and a variable `y`. Where is `y`, you say? `y` comes from evaluating the function, `f(x)`, at specific values of `x`; `y` is the output of the function. So `y` is `f(x)`. Sometimes you will see just the `f` without the `(x)` part. Don't think that's something different; it's not. The `f` by itself refers to the function in general; the set of operations that are performed on the input to the function, whatever that input happens to be. `f(x)` refers to an instance of the function, `f`. A function has to have an input, and the input is whatever is inside the parentheses. Let's go through a couple of examples and hopefully, this will make more sense.

Suppose you have a function, `f(x)=x^2 + 2x + 1`. This takes `x` as its input (what's inside the parentheses) and the right-hand side of the equation defines what the function does to the input, in this case, `x`. When you see this function, you should think:

- square the input to the function,
- then add 2 times the input to that result,
- and finally add 1 to that result

Now consider the function, `g(x)=sqrt(4x+7)`. Again the input to this function, `g`, is just `x`. With this function you should think:

- multiply the input to the function by 4,
- then add 7 to that result,
- and finally take the square root of that result

Another confusing function thing for many people is when we have an instance of a function like, `f(x+1)`. In general, this is not equal to `f(x) + 1`. Consider the function in the first example above, `f(x) = x^2 + 2x + 1`. Remember that whatever's inside the parentheses in the function notation defines what the input to the function is, and the right-hand side of the equation defines what the function does to the input. It might be clearer to write the function this way,

`f(text(input))=text(input)^2 + 2*text(input) + 1`

Now it should be clear that regardless of what the input is, you know what to do with it. So
`f(x) = x^2 + 2x + 1`

`f(x+1) = (x+1)^2 + 2(x+1) + 1`

`f(text(giraffe)) = (text(giraffe))^2 + 2*(text(giraffe)) + 1`

One more thing, what about a function like `f(x) = 3`? This function says to take the input, whatever it is, do nothing with it and just set the function value, which is also the corresponding `y` coordinate, to the constant value of `3`. That's it; it's a constant function. It doesn't matter what the input to the function is or what the value of `x` is, you always get `3` and only `3` as the output. It's graph is a horizontal line passing through `3` on the `y`-axis.
Consider the linear function, `f(x) = 5x + 3`. Here's what we know about this function just from the equation for `f(x)`.

- It's a linear (1st degree polynomial) function.
- Its graph is a line.
- The slope of the line is 5 (a constant).
^{}Remember the `y=mx+b` form of a line, where `m` is the slope. - The `y`-intercept of the function is 3 (i.e., the point `(0,3)` ), found by evaluating `f(0)`.
^{}Or also from the `y=mx+b` form of a line, where `b` is the `y`-intercept. - The `x`-intercept of the function is `-3/5` (i.e., the point `(-3/5,0)`) [aka, a zero of `f(x)`; the solution to the equation `f(x) = 0`]

Consider the quadratic function, `f(x)=3x^2 - 6x - 9`. Here's what we know about this function from the equation for `f(x)`.

- It's a quadratic (2nd degree polynomial)
^{}In this case 'quad' does not refer to `4`. The word 'quadratic' comes from the word, quadratum, the Latin word for square.function. - Its graph is a parabola that opens up.
^{}We know it opens up because the coefficient of the squared term is positive. - The slope of the parabola varies based on the value of `x`.
- The `y`-intercept of the function is `-9` (i.e., the point `(0,-9)` ), found by evaluating `f(0)`.
- The `x`-intercepts of the function are `3` and `-1` (i.e., the points `(3,0)` and `(-1,0)`) [aka, the zeroes of `f(x)`; the solution to the equation `f(x) = 0` solved by factoring, if possible, or using the quadratic formula]

Consider the cubic function, `f(x)=x^3 - 3x^2 - 33x + 35`. Here's what we know about this function from the equation for `f(x)`.

- It's a cubic (3rd degree polynomial)
^{}The word 'cubic' comes from the word, cubicus, the Latin word for cube ... it has `3` dimensions. function. - Its graph is a curve with at most two direction changes (one less than its degree).
- The slope of the function varies based on the value of `x`.
- The `y`-intercept of the function is `35` (i.e., the point `(0,35)` ), found by evaluating `f(0)`.
- The `x`-intercepts of the function are `-5`, `1`, and `7 ` (i.e., the points `(-5,0)`, `(1,0)`), and `(7,0)` [aka, the zeroes of `f(x)`; the solution to the equation `f(x) = 0`; solved in this course with a calculator]

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