Function Definitions

FUNCTIONS 12/6/12

Today we defined some important concepts regarding functions. To begin, remember that a function relates an input to an output, and there can only be one valid output for each input (or only one y value for x value). 

Below are some of the definitions we discussed in class with explanations

A function ƒ is increasing on an interval if, for any x₁ and x₂ in the interval, x₁<x₂ implies that    

In other words, your smaller x input value will always give you your smaller y output. This will cause the graph of an increasing equation to have a positive slope within your interval.

A function ƒ is decreasing on an interval if, for any x₁ and x₂ in the interval, x₁<x₂ implies that    

Basically, this means the opposite of the previous definition. In simpler words, it means that for your smaller x input, you will get your larger y output. This would mean that the graph of a decreasing function would have a negative slope on your interval. 

A function ƒ(a) is called a relative maximum of ƒ if there exists an interval (X₁, X₂) that contains a such that 

 implies 

To better understand this definition, you can think of it like f(a) > f(anything else on the interval). On a graph of a relative maximum function like this, you would see a parabola opening downward with a maximum. This maximum would be at the coordinate (a, f(a)). So by inputting the value a, you will get the greatest possible output. This maximum point is also known as the extremum. 

A function ƒ(a) is called a relative minimum of ƒ if there exists an interval (X₁, X₂) that contains a such that 

   implies

An easy way to think about relative minimums is to remember that f(a) < f(anything else on the interval). So, when you input a, you will get your least possible output value f(a). The minimum point on the parabola (which opens upward) is also called the extremum. 

With all of the above definitions, it is important to remember that they only refer to your interval. The graph of a function can change and do crazy things outside of your interval, but you only need to look at the chunk of the graph that's in your interval. 

A function f is even if, for each x in the domain of f,

f (-x) = f (x) 

So basically, when you plug -x into your function, you get the same output that you would get by just plugging in x. By looking at a graph, you can recognize that a function is even. Even functions have symmetry about the y axis. For example, a parabola centered on the y axis would be an even function because each side is reflected over the axis. 

A function f is odd if, for each x in the domain of f, 

f (-x) = - f (x)

In other words, when you plug -x into your function, your output will be the same as -1( f (x) ). You can recognize an odd funciton on a graph because they appear to be reflected about the origin, like the graph below. 

Functions are super important so make sure you understand this stuff. Alright well that's it. Just remember these definitions when you're tryna function. 

 

-Corn 

oh and heres a picture of a young Mr. Wilhelm looking dapper as always. Google works wonders.