Functions and Function Notation
Description/Explanation/Highlights
Video Description
This video explains what a function is, how to identify a function, and how to use function notation.
Steps and Key Points to Remember
To define and identify a function and to write a function in function notation, follow these steps:
- A function is defined as a relation that for any independent variable there is one and only one dependent variable associated with it.
- This means that for any x-value there is only one associated y-value in a function.
- It may be helpful to think that for any number you choose for x, there will be only one answer. This is an important piece in making functions useful. We need to be able to find one answer.
- In a function, it is acceptable for many x-values to be associated with the same y-value, but not the opposite.
- In a simple example, \(y=x+2\) plugging any number in for x will always get only one answer for y. This equation is therefore a function. For example if we plug 7 in for the x like this: \(y=(7)+2\), the value of y has only one answer: 9.
- A table of x- and y-values represents a function when every x-value represents only one y-value. If the x-values repeat and assign the same x-value to two different y-values, that table does not represent a function.
- A graph will represent a function if a vertical line drawn through any point on the graph will only intersect the graph in the one point. If a vertical line can be drawn anywhere (even if only in one place) on the graph that will intersect the graph in more than one point, that graph does not represent a function.
- As we saw in the example in #5 above, an equation can be used to represent a function symbolically if plugging in any value in the domain for x will get only one value for y.
- We can even represent functions verbally in “word problems.” For example if we say that an employee will be paid $15 per hour and ask how much an employee will be paid for any given number of hours work, we will be representing a function. For any number of hours (or partial hours) worked, we will be able to determine one and only one total pay amount.
- Once we know that a relation represented symbolically is a function, we often represent it using function notation.
- \(y=x^2+3\) written in function notation is \(f(x)=x^2+3\) and is read, “f of x equals x squared plus 3.” \(f(x)\) means the same thing as y.
- Function notation does not require us to use the letter “f” but allows us to choose any letter to “name” the function. For example: \(g(x)\), \(m(x)\), or \(p(x)\). That can be valuable in identifying individual functions when referring to them.
- Function notation makes it easy to evaluate for a value of x. For example instead of saying, “What is the value of y when x is 4 in the equation \(y=3x-5\), we can write \(f(x)=3x-5\) and ask, “What is the \(f(4)\)?” Which is read, “What is the f of 4?” Both statements tell us to substitute 4 for the value of x in the equation but in function notation, once the function is written and labeled “f” we can simply refer back to it when asking for another value to be substituted. For example: \(f(19)\) tells us to substitute 19 for x in function f.
- In the example \(f(x)=x^2+3\), find \(f(1)\) simply means substitute 1 for x in function f so \(f(1)=(1)^2+3=4\).
Here are some key points to keep in mind when identifying functions and working with function notation.
- In all forms of function representation, an x-value will always be associated with only one y-value.
- Any value of x in the function can get only one answer when evaluated for y,
- For a table to represent a function, the x-values will not repeat (unless they all assign the same value for y).
- Y-values can repeat.
X
-2
1
5
7
Y
2
3
4
4
The table of x- and y-values to the left represents a function because the x-values do not repeat. Notice that the y-value of 4 is used twice. This is allowed. It simply assigns the x-value of 5 to y = 4 and the x-value of 7 to y = 4 as well. No x-value is assigned to more than one y-value.
X
-2
1
5
7
7
Y
2
3
4
4
2
The table of x- and y-values to the left does not represent a function because the x-value of 7 repeats and assigns a different value of y to 7 each time. The value of y when evaluated at 7 is either 4 or 2. This is not allowed in a function.
- Graphs of potential functions should be checked with the vertical line test. If a graph represents a function, there should be no vertical line that can be drawn that will pass through more than one point on the graph.
- Equations will represent functions as long as evaluating them for any value of x in the domain will result in only on y for the answer.
- Functions can be represented symbolically by using function notation.
- Letters are used to name functions in function notation and are written as \(f(x), g(x), m(x)\) etc.
- \(g(4)\) would mean to evaluate the g function by substituting 4 for the x-values in the function.
Video Highlights
- 00:00 Introduction
- 00:14 Definition of a function
- 00:37 \(y=x+2\) example of a symbolic function
- 01:12 Representation of a function using a table
- 03:38 Representation of a function as a graph
- 04:40 Graphs of functions and the vertical line test
- 05:10 Representing functions symbolically
- 05:52 Verbal representation of functions
- 06:42 Using function notation
- 07:20 \(f(1)=(1)^2+3=4\) example of evaluating a function
- 08:36 Conclusion
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