Algebraic Functions Versus Relations | Restricted Domain: Finding the Domain Algebraically |

Vertical Line Test | More Practice |

Domain and Range of Relations and Functions |

**Note**: More **advanced topics with functions** can be found in the **Parent Functions and Transformations** and **Advanced Functions: Compositions, Even and Odd, and Extrema** section. **End behavior** of functions can be found here in the **Parent Functions and Transformations** section, here in the **Graphing and Finding Roots of Polynomial Functions** section, and also in the **Graphing Rational Functions, including Asymptotes** section. **Intermediate Value Theorem (IVT)** can be found here in the **Limits and Continuity** section.

## Algebraic Functions Versus Relations

When we first talked about the coordinate system, we worked with the graph that shows the relationship between how many hours we worked (the independent variable, or the “$ x$”), and how much money we made (the dependent variable, or the “$ y$”). Any relationship between two variables, where one depends on the other, is called a **relation**, since it relates two things.

This particular relation is an **algebraic** **function**, since there is only one $ y$ for each $ x$. In other words, since the $ x$ is the “question” and $ y$ is the “answer”, we can only have one answer for each question. For whatever is the number of hours we work, we only get paid a certain amount for that: $ y=10x$.

Again, a function is just a fancy way of saying something depends on something else, and there’s only **one** “$ y$” **for every** “$ x$”. Functions are typically written with $ f\left( x \right)$ instead of $ y$, so we can write this function $ f(x)=10x$. Note that this is **not “**$ f$** times **$ x$**”**; it is “$ f$** of **$ x$”. What it means is that $ x$ is on the **right-hand sign** of the “=” sign, and you can put different values in for $ x$ on the left-hand side to get one and only one value on the **right-hand sign**. So again, “$ f(x)$” is really “$ y$”. It’s that simple. Here are some examples:

Again, what makes a relation a function is that you can only have one “answer” (the $ y$) for each “question” (the $ x$). **All functions are relations, but not all relations are functions.**

## Vertical Line Test

Notice that when we have a function, we can’t draw a vertical that goes through more than one point. This is called the **vertical line test**, and it’s a useful tool to determine if a graph is a function or not. For example, we can tell the following graph is not a function since we can draw a vertical line and hit more than one point:

Here are some examples of functions. Notice that you **can have** two “questions” ($ x$) for the same “answer” ($ y$):

Another way to look at a set of points and determine whether or not they are functions is to draw what we call **mapping diagrams**, since we are mapping the $ x$-values to the $ y$-values. We order values from smallest to largest and don’t repeat the values on each side and match them up. If we have more than one $ y$-value for one $ x$-value, we don’t have a function. Here are some examples:

## Domain and Range of Relations and Functions

**Domain** and **Range** of functions (and relations) sound really difficult and scary, but they are not really bad at all. You know how those mathematicians like to use fancy words for easy stuff?

Remember that since “**d**” comes before “**r**”, the **domain **of functions has to do with the “$ x$”’s and the **range** of functions have to do with the “$ y$”’s. To get the domain, we are just looking for all the **possible** values of $ x$ **for that function **(from smallest to largest), and for the range, we are looking for all possible values of $ y$ **for that function **(again, from smallest to largest).

To help me do this, I like to use my pencil – but it’s backwards compared to what you might think. To find the domain, I put my pencil **vertically** and start at the **left** and see where it first hits a point. Then I push it through all the way to the right to see where it ends hitting points. For the range, I do the same thing, but with a **horizontal** pencil that’s moving **up**:

Here are more examples, using what we call “**Interval Notation**”. (We saw this in the **Linear Inequalities** Section). This is the most commonly used way to describe domains and ranges, and it always goes from lowest to highest with “**(**“ (**soft brackets**) if the relation or function doesn’t hit the point (including $ \infty $, and $ -\infty $), and “**[**“ (**hard brackets**) if the relation or function does hit the line. If you have to skip over any numbers, you do so by using the “**U**” sign, which means **union**, or putting things together.

We can also use **Set Builder**/**Inequality Notation**, where, as we saw before, we use inequality signs to describe the ranges.

**Notice that when we see arrows in the graphs, we have to assume that the function “goes on forever” in those directions. ****Remember that closed circles means the function includes that point, and open circles means the function doesn’t include that point.**

If you don’t see how we got the domain and range above, use the pencil trick, and make sure you start from the left for the domain (with vertical pencil) and from the bottom with the range (with horizontal pencil).

## Restricted Domains: Finding the Domain Algebraically

In most cases, the **domain is restricted**:

- It is
**randomly indicated**that way in the problem. For example, $ f\left( x \right)=3x-1,\,\,x\ge 0$. - There is a
**variable in the denominator**and that denominator could be**0**. For example, $ \displaystyle f\left( x \right)=\frac{1}{{x-3}}$. (In this case, $ x-3\ne 0;\,\,\,x\ne 3$). - There is a
**variable underneath an even radical sign**, and that radicand (underneath the radical sign) could be negative. For example, $ f\left( x \right)=\sqrt{{x+4}}$. (In this case, $ x+4\ge 0;\,\,\,\,x\ge -4$). - (More advanced – see
**Logarithmic Functions**section) If there’s a**variable in the argument of a log**or**ln**function; log arguments must be greater than**0**. For example, $ f\left( x \right)=\log \left( {8-x} \right)$. (In this case, $ 8-x>0;\,\,\,x<8$).

There are other types of functions, like **Trigonometric Functions**, that have domain restrictions, but we won’t address these here.

Note that if we could have a mixture of the above restrictions, for example, for $ \displaystyle f\left( x \right)=\frac{{\sqrt{x}}}{{x-1}}$, $ x\ge 0$ and $ x\ne 1$, so the domain of $ x$ is $ \left[ {0,1} \right)\cup \left( {1,\infty } \right)$.

We start out assuming that the **domain of a function is all real numbers**, but then see **if there are any exceptions**, as seen in the table. We will learn more about **rational functions** (shown in the first two examples, where there are variables in the denominator) in the** Rational Functions and Equations**, and **Graphing Rational Functions, including Asymptotes** sections. Also, domains of composite functions information is in the **Advanced Functions** section here. Here are more detailed examples:

We will work on more advanced topics with functions later, in the **Advanced Functions** section.

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On to **Advanced Functions: Compositions, Even and Odd, and Extrema** you are ready!