A function is a connection between an independent variable $(X)$ and a dependent variable $(Y)$. The relationship between the variables is called a "correspondence rule".

An algebraic representation of a function is actually a description of the relationship between the dependent variable $(Y)$ and the independent variable $(X)$ by means of an equation.

The following is the typical structure of a graphical representation:

• $Y=X+3$, $Y=2X-5$

For example, if the data is that every month, Daniel earns$20.000$ dollars.

The algebraic representation will be $X$ for the number of months $Y$ $f (X)$ for the amount earned. $f (x) = 20000X$

## Examples with solutions for Algebraic Representation of a Function

### Exercise #1

Is the given graph a function?

### Step-by-Step Solution

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

Let's note that in the graph:

$f(0)=2,f(0)=-2$

In other words, there are two values for the same number.

Therefore, the graph is not a function.

No

### Exercise #2

Is the given graph a function?

### Step-by-Step Solution

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

We should note that for every X value found on the graph, there is one and only one corresponding Y value.

Therefore, the graph is indeed a function.

Yes

### Exercise #3

Is the given graph a function?

### Step-by-Step Solution

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

We should note that for every X value found in the graph, there is one and only one corresponding Y value.

Therefore, the graph is indeed a function.

Yes

### Exercise #4

Determine whether the following table represents a function

### Step-by-Step Solution

It is important to remember that a constant function describes a situation where as the X value increases, the function value (Y) remains constant.

In the table, we can see that there is a constant change in the X values, specifically an increase of 2, and the Y value remains constant.

Therefore, according to the rule, the table describes a constant function.

Yes

### Exercise #5

Which of the following equations corresponds to the function represented in the graph?

### Step-by-Step Solution

Use the formula for finding slope:

$m=\frac{y_2-y_1}{x_2-x_1}$

We take the points:

$(0,-2),(-2,0)$

$m=\frac{-2-0}{0-(-2)}=$

$\frac{-2}{0+2}=$

$\frac{-2}{2}=-1$

We substitute the point and slope into the line equation:

$y=mx+b$

$0=-1\times(-2)+b$

$0=2+b$

We combine like terms:

$0+(-2)=b$

$-2=b$

Therefore, the equation will be:

$y=-x-2$

$y=-x-2$

### Exercise #6

Which of the following equations corresponds to the function represented in the table?

### Step-by-Step Solution

We will use the formula for finding slope:

$m=\frac{y_2-y_1}{x_2-x_1}$

Let's take the points:

$(-1,4),(3,8)$

$m=\frac{8-4}{3-(-1)}=$

$\frac{8-4}{3+1}=$

$\frac{4}{4}=1$

We'll substitute the point and slope into the line equation:

$y=mx+b$

$8=1\times3+b$

$8=3+b$

Let's combine like terms:

$8-3=b$

$5=b$

Therefore, the equation will be:

$y=x+5$

$y=x+5$

### Exercise #7

Is the given graph a function?

No

### Exercise #8

Is the given graph a function?

No

### Exercise #9

Determine whether the following table represents a function

No

### Exercise #10

Determine whether the data in the following table represent a constant function

No

### Exercise #11

Is the given graph a function?

Yes

### Exercise #12

Is the given graph a function?

Yes

### Exercise #13

Determine whether the following table represents a function

Yes

### Exercise #14

Determine whether the following table represents a function

Yes

### Exercise #15

Determine whether the following table represents a function