~Extremely important in the study of higher academic disciplines.  Widely used in calculus & calculus related courses.

 

~The equations of functions may be represented several ways in problem solving. Let me list the popular ways using the equation y=x². The dependent variable is y and the independent variable is x. The value of the dependent variable "depends" on the value of the independent variable substituted into the equation.

 

~Graphically speaking, the values of the dependent variable are usually plotted on the vertical axis and those of the independent variable on the horizontal axis.  We say, "y is a function of x". The graph gives an illustration of all the ordered pairs (x,y) described by the equation. Functions have the property that all vertical lines that intersect the graph, must do so only once.

This property is called the "vertical line test" for a functional graph.

 

~y=x² is one way of representing this function. This is called the dependent variable notation. This way is not always convenient in problem solving.

 

~Substituting f(x) for y gives, f(x)=x². This is the functional notation for the same function. The meaning of f(x) is not multiplication (i.e., f times x), instead it stands for the phrase, "function of x".  So y=f(x) simply says, "y is a function of x". We place the independent variable, in this case, x, inside those parenthesis. The dependent & independent variables could be a number of different quantities, depending on the problem we are analyizing.

 

~Since functional notation is used quite extensively through out many courses, it is extremely important to know how to use this notation properly. Let me give examples of using it with the simple function, y=x² or f(x)=x²

 

~To find values of y on this curve, we would substitute an x value into the equation. For example, for x=0, y=0²=0.  The equivalent functional notation would look like this:  f(0)=0, much nicer. So, a functional value is nothing more than a y value when the value of the independent variable (x in this case) is placed within the parenthesis. Let me give you more values & expressions using this function.

 

f(1)=1²=1

f(-1)=(-1)²=1

f(-3)=(-3)²=9

f(x²)=(x²)²=x⁴

f(x+h)=(x+h)²=x²+2hx+h² (the meaning of this expression will be explained later)

 

~So, whatever we place between the parenthesis, we operate on it using the function we are dealing with..

 

~Here are more examples using a variety of different functions:

 

If f(t)=3-2t-3t⁴, then f(-2)=3-2(-2)-3(-2)⁴= -41

 

If g(p)=p/(1+p), then g(2)=2/(1+2) = 2/3

 

If f(x)=3x-1, then f(x+h)-f(x-h) = 3(x+h)-1-[3(x-h)-1]=3x+3h-1-[3x-3h-1]= 6h

This one is tricky with those minus signs & parenthesis...be careful here..

 

~Some teachers combine both notations. For example, for y=x², some use,

y(x)=x²....so, y(0)=0 would mean, "the y value when x=0 is 0".

 

~In conclusion, practice & more practice is probably the only way of becoming proficient using functional notation.