## Lesson 2: Functions and Function Notation

Functions are written using function notation. To write a function using function notation, substitute the dependent variable with f(x). f (any letter can be used) is used to denote a function and x represents the independent variable in the function. The value of the independent variable being used in a certain case replaces x. So f(4) is the value of a function when the independent variable 4 is used. Let's look at an example:

y= 3x+10

First, replace the dependent variable, or y in this case with f(x):

f(x)= 3x+10

Now we can find the value of the function for any value of x. What is the value of the function when x=3? Simply substitute 3 for every x in the function. Note that the three in the brackets means that the independent variable 3 is being used.

f(x)=3x+10

f(3)=3(3)+10 (Substitute x=3)

f(3)=9+10

f(3)=19

As a result, you can find the value of any function for any independent variable by simply substituting the value for x in function notation. When a relation is given as a set of ordered pairs, the information can be organized into a mapping diagram. In a mapping diagram, domain values are placed in one oval and range values in an adjacent oval. Arrows are then drawn from the domain values to their corresponding range values. A relation represented through a mapping diagram is a function if only one arrow leaves each value in the domain. If two or more values leave any value in the domain, the relation is not a function; a value in the domain has more than one corresponding value in the range. Let's look at a mapping diagram and interpret it:

y= 3x+10

First, replace the dependent variable, or y in this case with f(x):

f(x)= 3x+10

Now we can find the value of the function for any value of x. What is the value of the function when x=3? Simply substitute 3 for every x in the function. Note that the three in the brackets means that the independent variable 3 is being used.

f(x)=3x+10

f(3)=3(3)+10 (Substitute x=3)

f(3)=9+10

f(3)=19

As a result, you can find the value of any function for any independent variable by simply substituting the value for x in function notation. When a relation is given as a set of ordered pairs, the information can be organized into a mapping diagram. In a mapping diagram, domain values are placed in one oval and range values in an adjacent oval. Arrows are then drawn from the domain values to their corresponding range values. A relation represented through a mapping diagram is a function if only one arrow leaves each value in the domain. If two or more values leave any value in the domain, the relation is not a function; a value in the domain has more than one corresponding value in the range. Let's look at a mapping diagram and interpret it:

Since the domain values x=-1 and x=1 corresponds with more than one value in the range, the relation is not a function. The set of ordered pairs proves this:

{(-1,3), (-1,4), (1,1), (1,2), (2,3), (4,3)}

Since the x values x= -1 and x=1 correspond with two y-values, the relation is not a function.

Function notation can also be written in another way: mapping notation. To write a function in mapping notation simply replace f(x) with f:x. Also replace the equal sign with (→) For example:

f(x)= 2x+3 can be written in mapping notation as:

f:x→2x+3

Function notation can be used to solve word problems. Let's look at a word problem:

The temperature at the surface of the Adriatic sea is 24°C on a hot summer day. A

researcher discovers that for every 5m he descends the temperature drops 3°C.

What is the temperature at a depth of 15m?

First, model the water temperature using function notation. Then substitute 15, the depth, for the independent variable to find the dependent variable- the

temperature:

Let d represent the depth, in metres, and let f represent the temperature, in °C.

f(d)= 24-3(d/5)

f(15)= 24-3(15/5) Substitute x=15

f(15)=24-3(3)

f(15)=24-9

f(15)=15

The temperature at a depth of 15 m is 15°C (don't forget to write a concluding sentence).

{(-1,3), (-1,4), (1,1), (1,2), (2,3), (4,3)}

Since the x values x= -1 and x=1 correspond with two y-values, the relation is not a function.

Function notation can also be written in another way: mapping notation. To write a function in mapping notation simply replace f(x) with f:x. Also replace the equal sign with (→) For example:

f(x)= 2x+3 can be written in mapping notation as:

f:x→2x+3

Function notation can be used to solve word problems. Let's look at a word problem:

The temperature at the surface of the Adriatic sea is 24°C on a hot summer day. A

researcher discovers that for every 5m he descends the temperature drops 3°C.

What is the temperature at a depth of 15m?

First, model the water temperature using function notation. Then substitute 15, the depth, for the independent variable to find the dependent variable- the

temperature:

Let d represent the depth, in metres, and let f represent the temperature, in °C.

f(d)= 24-3(d/5)

f(15)= 24-3(15/5) Substitute x=15

f(15)=24-3(3)

f(15)=24-9

f(15)=15

The temperature at a depth of 15 m is 15°C (don't forget to write a concluding sentence).

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