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An Introduction to Functions

Sep-22-2021 | Hits: 282 |

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:

1. square the input to the function,
2. then add 2 times the input to that result,
3. 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:

1. multiply the input to the function by 4,
2. then add 7 to that result,
3. and finally take the square root of that result

Remember to start inside parentheses or inside square roots and work your way out from there.

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,

Now it should be clear that regardless of what the input is, you know what to do with it. So 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.

Linear Polynomial Functions

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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`]

Quadratic Polynomial Functions

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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]

Cubic Polynomial Functions

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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]