# Linear Functions

Hi, and welcome to this video on Linear Functions! In this video, we will focus on **linear functions**, which are algebraic, and we will discuss the difference between a linear function and a linear equation.

Before we jump right into linear functions, it is important to review some basic terminology:

A **polynomial expression** is comprised of terms joined by addition or subtraction. A **term** can be a constant, a variable, or a constant multiplied by one or more variables. In addition, variables within polynomials are raised to positive **exponents**. Here are a few examples of two, three, and four term polynomials:

The **degree** of a single variable polynomial expression is determined by the largest exponent of all terms. A **linear** expression is defined as having the **degree** equal to **one**.

Note that these polynomial expressions do not have an equal sign, but are simply terms added or subtracted. An **equation** includes an equal sign and, as a result, relates two variables according to the expression.

In other words, y can be written in terms of x. For example:

y equals 3x is saying that y is triple the size of x.

y equals x plus 3 is saying that y is three more than x.

And y equals 2x minus 4 is saying that y is four less than twice the value of x.

Equation | Translation |

\(y=3x\) | “\(y\) is triple the size of \(x\)” |

\(y=x+3\) | “\(y\) is three more than \(x\)” |

\(y=2x-4\) | “\(y\) is four less than twice the value of \(x\)” |

As you can imagine, many real-life scenarios can be translated into equations once variables are defined. The equation simply establishes the **relationship** between variables.

**Linear functions** are similar in that they have an equal sign; however, function notation has the unique meaning of “output.” The most common example of function notation is \(f\) of \(x\). This means that the function, named \(f\), is expressed in terms of the variable, \(x\). The function is then evaluated at values of \(x\) for the desired output.

The equations we mentioned before can be expressed as functions in terms of the variable \(x\) as follows:

The function \(d\) of \(x\) equals \(3x\).

The function \(g\) of \(x\) equals \(x\) plus 3.

And the function \(h\) of \(x\) equals \(2x\) minus 4.

**Function**

\(d(x)=3x\)

\(g(x)=x+3 \)

\(h(x)=2x-4\)

It is very important that the variable is consistent within the function notation and the expression. When it’s time to evaluate the function, the variable will be replaced with a particular value. While there are no rules with regard to the naming of functions, it is sometimes useful to name the function according to what it represents.

For example, a function that represents the total cost of renting a car where the variable, \(x\), represents the number of days may be written as \(C\) of \(x\) equals \(36x\) plus 8, where the daily charge is $36 and the upfront cost of renting is $8.

The function notation for a 4-day rental would be \(C\) of 4, “the cost of renting for 4 days”

To find the cost of renting the car for 4 days, simply substitute the \(x\) for 4 and evaluate the expression:

\(C(4)=152\)

When evaluating functions, it is important to interpret in the appropriate context.

“The cost of renting the car for four days is $152.”

Keep in mind that while both linear equations and linear functions may look similar, the notation defines the subtle difference. Equations, indicated by “\(y\)=”, represent how the variables are algebraically related. Function notation, \(f(x)\), enables evaluation at a specific value for a desired “output.” With practice, you will begin to recognize the distinctions and use the appropriate representation.

Thanks for watching, and happy studying!

## Practice Questions

**Question #1:**

Which of the following shows the function “*f* of *x* is equal to two more than three times *x*” in numeric form?

**Answer:**

The translation of the verbal description of the function into the numeric form of the function can be done by analyzing each part. The first part gives the name of the function, *f* of *x*. The next part says, “is equal to two more than…” which means two is being added. The last part says, “more than three times *x*”, which means 3 and *x* are being multiplied. Putting all this together gives us: \(f(x)=3x+2\).

**Question #2:**

Essa pays a monthly membership fee of $5 to an online video game shop to be able to download video games at a discount rate of $3 per game. Which function, \(v(x)\), can Essa use to calculate the total monthly bill for downloading *x* number of video games?

**Answer:**

In the function, \(v(x)=3x+5\), the constant (5) is the one-time monthly membership fee. Since *x* represents the number of video games and each video game is $3, the term \(3x\) shows the cost of the total number of video games bought for the month. Adding these two things together will give us her total monthly bill.

**Question #3:**

Hudson receives a flat fee of $100 plus $20 for each lawn, *x*, that he mows. The function, \(p(x)\), represents Hudson’s monthly salary for mowing lawns. Which function can Hudson use to calculate his monthly salary for mowing *x* number of lawns?

**Answer:**

The flat rate, $100, is represented by the constant. Since *x* represents the number of lawns that Hudson mows and $20 represents the amount he is paid for each lawn, the term \(20x\) gives the total amount he is paid for the total number of laws mowed in one month. Therefore, Hudson can use the function \(p(x)=20x+100\) to calculate his monthly salary.

**Question #4:**

Which is a correct verbal description of \(g(x)=\frac{1}{3}x-6\)?

The function, *g* of *x*, is one-third of six-time *x*.

The function, *g* of *x*, is six times one-third of *x*.

The function, *g* of *x*, is six less than one-third of *x*.

The function, *g* of *x*, is six more than one-third of *x*.

**Answer:**

When translating from a numeric to a verbal description, we will look closely at what is happening to each term. The first terms, \(\frac{1}{3}x\), describes what will happen to the value of *x*, which is multiplied by \(\frac{1}{3}\) or divided by 3, which translates to one-third of the value of *x*. Since 6 is being subtracted, we know whatever the value of one-third of *x* is, the value of the function *g* of *x* will be 6 less. Therefore, the verbal description, the function, *g* of *x*, is six less than one-third of *x*, accurately describes the function.

**Question #5:**

Which situation can be depicted by the function of \(f(x)=0.20x+2\)?

*f* of *x* is the total printing fee, where there is a $20 flat fee plus $2 per page, *x*.

*f* of *x* is the total printing fee, where there is a $2 flat fee plus $2 per page, *x*.

*f* of *x* is the total printing fee, where there is a $2 flat fee plus 20 cents per page, *x*.

*f* of *x* is the total printing fee, where there is a 20-cent flat fee plus $2 per page, *x*.

**Answer:**

A closer look at the function, \(f(x)=0.20x+2\), shows a constant of 2, which would tell us that this is a value that is only added one time and not a rate. The term \(0.20x\) tells us the rate is 0.20 for whatever the value of *x* represents. Therefore, the scenario, *f* of *x* is the total printing fee, where there is a $2 flat fee plus 20 cents per page, *x*, would be a possible situation that can be represented by this function.