GED Mathematical Reasoning: Introduction To Functions

  • A set of ordered pairs is called a relation where the x-values make up the “domain” of the relation and the y-values make up the “range.”
  • A function is a special kind of relation where each domain value (or x-value) corresponds to one and only one range value (or y-value).

 

Function notation 

In the past, I might have said or written these instructions:  Find y when x = -2 [/latex]

Now, we can use function notation to give the same directions by simply stating: Find f(-2). So to find f(-2), we’ll substitute -2 for x and simplify the right side of the equal sign to determine the corresponding y value. We’re not doing anything new here – it’s just that the notation is new.

For example, f(x) = 2x^2 - 7 is the same thing as  y = 2x^2 - 7

 

Example 1

Is the relation a function? Explain why or why not.

graph3

 

 

 

 

 

The relation shown in example one is composed of 5 ordered pairs:

(-2, 4), (-1, 1), (0,0), (1, 1) And (2, 4)

The domain of this relation is made up of all the x-values:  -2, -1, 0, 1, 2

The range of this relation is made up of all the y-values: 4, 1, 0

Although the numbers 1 and 4 occur twice in the table, we will state them only once in the listing of the range. Also – it is common to enclose the domain and range in brackets as shown.

Domain: {-2, -1, 0, 1, 2}

Range: {4, 1, 0}

To determine if this relation is a function, we’ll ask ourselves: Does each x-value correspond to one and only one y-value? It may be helpful to draw arrows to represent each correspondence.

coord24

Although the x-values -2 and 2 both correspond to the y-value 4, they each correspond to one and only one y-value. In other words, it’s okay if more than one x-value corresponds to the SAME y-value.

So the answer is: Yes! This relation is a function because each x-value corresponds to one and only one y-value.

 

Example 2

Is the relation a function? Explain why or why not.

graph4

 

 

 

 

 

The relation shown in example two is composed of 5 ordered pairs:

(5, 3), (5, -3), (4, 2), (3, -2) And (2, 0)

The domain of this relation contains the x-values:  5, 4, 3 and 2

The range of this relation is made up of all the y-values:  3, -3, 2, -2 and 0.

Domain: {5, 4, 3, 2}

Range: {3, -3, 2, -2, 0}

To determine if this relation is a function, we’ll ask ourselves: Does each x-value correspond to one and only one y-value? Let’s use arrows to represent the correspondence.

coord25

The answer is: No. Do you see how the domain value of 5 matches to TWO DIFFERENT values in the range, 3 and -3? Therefore, this relation is NOT a function because the x-value 5 corresponds to more than one y-value.

In the “real world” functions don’t come in the form of a table. They come in the form of an equation. Many equations represent functions, meaning that the ordered pair solutions to the equation meet the definition of a function.

Consider the equation:  y = 2x + 1 and its graph as shown.

coord26

In this case, every solution to the equation or every point on the line meets the definition of a function. Each x-value corresponds to one and only one y-value.

math 246

When this is true, we sometimes name the equation with a letter like: f , g or h. And since this equation represents a function, y = 2x + 1 one can be rewritten using function notation as:

y = 2x + 1 \rightarrow f(x) = 2x + 1

Before we move on to the next example, I want to point out two very important things.

First, all this implies that y and function notation f(x) are interchangeable. When we see a notation like f(x) we can think of that as being the same thing as y.

Second, f(x) is not f times x. Simply put, the notation f(x) is a shorthand way to communicate that the function’s name is f and the variable used on the right side is x.

At this point, you may be wondering: Why do we have function notation? One reason is that it allows us to communicate more efficiently.

 

Example 3

Given: f(x) = 3x^2 - 10, find f(-2)

 

f(x) = 3x^2 - 10 is the same thing as:  y = 3x^2 - 10.

In the past, I might have said or written these instructions:  Find y when x = -2 [/latex] Now, we can use function notation to give the same directions by simply stating: Find f(-2).

When we let x = -2, we get f(-2) = 3{(-2)}^2 - 10

f(-2) = 3(4) - 10
f(-2) = 12 - 10
f(-2) = 2

So we would say that f(-2) = 2. Stated another way, when x = -2, y = 2. We can also write the result as the ordered pair:  (-2, 2)

 

Example 4

Given: f(x) = \frac{1}{2}x - 7, state the rate of change and the y-intercept (as an ordered pair).

 

Especially since you’ve just begun learning about functions, you may find yourself immediately replacing the f(x) notation with y because it’s more familiar. And that’s okay. As with most new things, function notation takes a little bit of getting used to.

So in example 4, we can re-write the equation:

f(x) = \frac{1}{2}x - 7 \rightarrow y = \frac{1}{2}x - 7

When we do so, we can see that we have a linear equation in slope-intercept form:

Slope-intercept form: y = mx + b

m = slope, b = y-coordinate of the y-intercept

Since this equation represents a linear equation which meets the definition of a function, we call it a “linear function.”

To review, slope refers to the steepness of a line. When it comes to linear functions, we often refer to “slope” as a “rate of change.” This makes sense since slope, by definition, is the RATIO of the CHANGE in y-values to the CHANGE in x-values.

Slope = Rate of Change

So going back to example 4 where we are being asked to state the rate of change and y-intercept, this is the same as stating the slope and y-intercept.

The slope or rate of change is the coefficient of x, which is one-half.

The y-coordinate of the y-intercept is represented by the b value tacked on to the end of the equation, which is -7. As an ordered pair, the y-intercept is:  (0, -7).

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