Introduction to Functions and Graphs
A function is a relationship in which one value depends upon another value. For example, if you are buying candy bars at a certain price, there is a relationship between the number of candy bars you buy and the amount of money you have to pay.
Think about functions as a machine—you put something into the machine, and it spits something back out. For example, when you enter quarters into a vending machine, you get a snack. (That is, unless it gets stuck on the ledge.) This is like a function. The input to the function was quarters, and the output of the function was a snack.
Basically, a function is a set of rules for using input and to produce output, and usually, this involves numbers.
Functions are written in the form beginning with the following symbol:
For example, consider the function f(x) = 8x – 2. If you are asked to find f(3), you simply substitute the 3 into the given function equation.
f(x) = 8x – 2
becomes
f(3) = 8(3) – 2
f(3) = 24 – 2 = 22
So, when x = 3, the value of the function is 22.
You could also imagine functions that take more than one number as their input, like f(x,y) = x + y. That means that if you give the function the numbers 7 and 4 as input, the function spits out the number 11 as output.
Function tables portray a relationship between two variables, such as an x and a y. It is your job to figure out exactly what that relationship is. Let's look at a function table:
Notice that some of the data was left out. Don't worry about that! You can still figure out what you need to do to the x in order to make it the y. You see that x = 1 corresponds to y = 4; x = 2 corresponds to y = 5; and x = 4 corresponds to y = 7. Did you spot the pattern? Our yvalue is just our xvalue plus 3.
Functions and Coordinate Grids
X and Y Axis
Okay, here's a quick review of the coordinate grid.
In a coordinate grid, the horizontal axis is the xaxis, and the vertical axis is the yaxis. The place where they meet is the point of origin. Using this system, you can place any point on the grid if you give it an xvalue and a yvalue, conventionally written as (x,y). If you have two or more points, you have a line.
Now think about a basic function. If you input an initial value x, you get an f(x) value. If you call f(x) the yvalue, you can see how a typical function can spit out a huge number of points that can then be graphed. Look back at f(x) = 8x – 2. If x = 2, y = 14. If x = 3, y = 22. So, we already have two points for this line: (2,14) and (3,22). Once you know two points of a function, you can draw a line connecting them. And guess what? You have now graphed a function!
The xvalues are known as the independent variables. The yvalues depend on the xvalues, so the yvalues are called the dependent variables.
Verticle Line Test
Potential functions must pass the vertical line test in order to be considered a function. The vertical line test is the following: Does a vertical line drawn through a graph of the potential function pass through only one point of the graph? If YES, then the vertical line passes through only one point, and the potential function is a function. If NO, then the vertical line passes through more than one point, and the potential function is not a function.
Horizontal Line Test
The horizontal line test can be used to determine how many different xvalues, or inputs, return the same f(x)value. Remember, a function cannot have one input return two or more outputs, but it can have more than one input return the same output. For example, the function f(x) = x^{2} is a function, because no xvalue can return two or more f(x) values, but more than one xvalue can return the same f(x)value. Both x = 2 and x = –2 make f(x) = 4. To find how many values make f(x) = 4, draw a horizontal line through the graph of the function where f(x), or y, = 4.
You can see that the line y = 4 crosses the graph of f(x) = x^{2} in exactly two places. Therefore, the horizontal line test proves that there are two values for x that make f(x) = 4.
Domain
All of the xvalues of a function, collectively, are called its domain. Sometimes there are xvalues that are outside of the domain, but these are the xvalues for which the function is not defined.
The function f(x) = 3x has a domain of all real numbers. Any real number can be substituted for x in the equation and the value of the function will be a real number.
The function f(x) = – 4 has a domain of all real numbers excluding 4. If x = 4, the value of the function would be , which is undefined. In a function, the values that make a part of the function undefined are the values that are NOT in the domain of the function.
What is the domain of the function f(x) = √x?
The square root of a negative number is undefined, so the value of x must not be less than 0. Therefore, the domain of the function is x ≥ 0.
Range
All of the solutions to f(x) are collectively called the range. Any values that f(x) cannot be equal to are said to be outside of the range.
As you just saw, the function f(x) = 3x has a domain of all real numbers. If any real number can be substituted for x, 3x can yield any real number. The range of this function is also all real numbers.
Although the domain of the function f(x) = is all real numbers excluding 4, the range of the function is all real numbers excluding 0, because no value for x can make f(x) = 0.
What is the range of the function f(x) = √x?
You already found the domain of the function to be x ≥ 0. For all values of x greater than or equal to 0, the function will return values greater than or equal to 0.
Nested Functions and Newly Defined Symbols
Nested Functions
Given the definitions of two functions, you can find the result of one function (given a value) and place it directly into another function. For example, if f(x) = 5x + 2 and g(x) = –2x, what is f(g(x)) when x = 3?
Begin with the innermost function: Find g(x) when x = 3. In other words, find g(3). Then, substitute the result of that function for x in f(x): g(3) = –2(3) = –6, f(–6) = 5(–6) + 2 = –30 + 2 = –28. Therefore, f(g(x)) = –28 when x = 3.
What is the value of g(f(x)) when x = 3?
Start with the innermost function—this time, it is f(x): f(3) = 5(3) + 2 = 15 + 2 = 17. Now, substitute 17 for x in g(x): g(17) = –2(17) = –34. When x = 3, f(g(x)) = –28 and g(f(x)) = –34.
Newly Defined Symbols
A symbol can be used to represent one or more operations. A symbol such as # may be given a certain definition, such as "m#n is equivalent to m^{2} + n." You may be asked to find the value of the function given the values of m and n, or you may be asked to find an expression that represents the function.
If m#n is equivalent to m^{2} + n, what is the value of m#n when m = 2 and n = –2?
Substitute the values of m and n into the definition of the symbol. The definition of the function states that the term before the # symbol should be squared and added to the term after the # symbol. When m = 2 and n = –2, m^{2} + n = (2)^{2} + (–2) = 4 – 2 = 2.
If m#n is equivalent to m^{2} + 2n, what is the value of n#m?
The definition of the function states that the term before the # symbol should be squared and added to twice the term after the # symbol. Therefore, the value of n#m = n^{2} + 2m. Watch your variables carefully. The definition of the function is given for m#n, but the question asks for the value of n#m.
If m#n is equivalent to m + 3n, what is the value of n#(m#n)?
Begin with the innermost function, m#n. The definition of the function states that the term before the # symbol should be added to three times the term after the # symbol. Therefore, the value of m#n = m + 3n. That expression, m + 3n, is now the term after the # symbol: n#(m + 3n). Look again at the definition of the function. Add the term before the # symbol to three times the term after the # symbol. Add n to three times (m + 3n): n + 3(m + 3n) = n + 3m + 9n = 3m + 10n.
Slope and yIntercept
Graphing Linear Equations
A linear equation always graphs into a straight line. The variable in a linear equation cannot contain an exponent greater than one. It cannot have a variable in the denominator, and the variables cannot be multiplied.
The graph of a linear equation is a line, which means it goes on forever in both directions. A graph is a picture of all the answers to the equation, so there is an infinite (endless) number of solutions. Every point on that line is a solution.
There are several methods that can be used to graph linear equations; however, you will use the slopeintercept method here.
Slope of a Line
What does slope mean to you? If you are a skier, you might think of a ski slope. The slope of a line has a similar meaning. The slope of a line is the steepness of a line. What is the yintercept? Intercept means to cross, so the yintercept is where the line crosses the yaxis. A positive slope will always rise from left to right. A negative slope will always fall from left to right.
To graph a linear equation, you will first change its equation into slopeintercept form. The slopeintercept form of a linear equation is y = mx + b, also known as y = form. Linear equations have two variables. For example, in the equation y = mx + b, the two variables are x and y. The m represents a number and is the slope of the line and is also a constant. The b represents a number and is the yintercept. For example, in the equation y = 2x + 3, the number 2 is the m, which is the slope. The 3 is the b, which is the yintercept. In the equation y = –3x + 5, the m is –3, and the b is 5.
Rise Over Run
Okay, now you know that slope means the steepness of a line. In the equation y = 2x + 3, the slope of the line is 2. What does it mean when you have a slope of 2? Slope is defined as the rise of the line over the run of the line. If the slope is 2, this means , so the rise is 2 and the run is 1.
If the slope of a line is , the rise is 2 and the run is 3. What do rise and run mean? Rise is the vertical change, and run is the horizontal change. To graph a line passing through the origin with a slope of , start at the origin. The rise is 2, so from the origin, go up 2 and to the right 3. Then draw a line from the origin to the endpoint. The line you have drawn has a slope of .
Now draw a line with a slope of . Start at the origin. Go down 3 units because you have a negative slope. Then go right 4 units. Finally, draw a line from the origin to the endpoint. These two lines appear on the same graph that follows.
TIP: Always look at all coordinate planes carefully to see what scale each is drawn to—some may have increments of one, while others may have increments of onehalf or ten.
Let's try another one. To graph an equation like y = x + 1, you can use the slope and yintercept. The first step is to figure out what the slope is. The slope is the number in front of x, which means in this case that it is 1. What is the yintercept? It is also 1. To graph the equation, your starting point will be the yintercept, which is 1. From the yintercept, use the slope, which is also 1, or . The slope tells you to go up 1 and to the right 1. A line is drawn from the yintercept to the endpoint (1,2). You can extend this line and draw arrows on each end to show that the line extends infinitely.
Find the Slope of an Equation  Getting the Right Form
What if the equation is not in slopeintercept form? Simple! All you need to do is change the equation to slopeintercept form. How? Slopeintercept form is y = form, so your strategy is to get the y on a side by itself.
An equation needs to be in slopeintercept form, or y = form (y = mx + b), before you can graph the equation with a pencil and graph paper. Also, if you use a graphing calculator to graph a linear equation, the equation needs to be in y = form before it can be entered into the calculator.
Example
2x + y = 5 
Subtract 2x from both sides of the equation. 
2x – 2x + y = 5 – 2x 
Simplify. 
y = 5 – 2x 
Rearrange the equation so the x term is first. 
y = –2x + 5 
Find the Slope of an Equation  Using the Commutative Property
There is a mathematical rule called the commutative property that lets you change the order of numbers or terms when you add or multiply. You want the preceding equation in the form y = mx+ b, so the order of the 5 and the –2x needs to be changed after getting the y on a side by itself. When you move a term, be sure to take the sign of the term with it. For example, the 5 was a positive number in the original order. It remains a positive number when you move it.
Example
2x + 3y = 9 
Subtract 2x from both sides of the equation. 
2x – 2x + 3y = 9 – 2x 
Simplify 
3y = 9 – 2x 
Use the commutative property. 
3y = –2x + 9 
Divide both sides by 3. 

Simplify both sides of the equation. 

Tip: Look at the last two steps of the 2x + 3y = 9 example. Because and x name the same number, you may see it written either way.
Example
–3x + 2y = 10 
Add 3x to both sides of the equation. 
–3x + 3x + 2y = 10 + 3x 
Simplify. 
2y = 10 + 3x 
Use the commutative property. 
2y = 3x + 10 
Divide both sides of the equation by 2. 

Simplify both sides of the equation. 

Linear Inequalities
The Deal with Linear Inequalities
A linear inequality has two variables just like a linear equation. The inequality 2x + y < 1 is a linear inequality with two variables. You can draw on what you already know to graph linear inequalities. A linear equation graphs into a line. A linear inequality has two parts: a line and a shaded area.
When you graphed linear equations, your first step was to put the equation into y = form. Do the same with the linear inequality. The commutative property lets you change the order of numbers or terms when you add or multiply. When you move a term, be sure to take the sign of the term with it.
Tip: If the inequality symbol is < or >, the boundary line will be dotted. If the inequality symbol is ≤ or ≥, the boundary line will be solid.
Example
2x + y > 1 
Subtract 2x from both sides of the inequality. 
2x – 2x + y > 1 – 2x 
Simplify. 
y > 1 – 2x 
Use the commutative property. 
y > –2x + 1 
The inequality y > –2x + 1 tells you that the slope is –2 and the yintercept is 1. If the inequality has a > or < symbol, then the line will be dotted. If the inequality symbol is ≤ or ≥, then the line will be solid. Generally, if the inequality symbol is > or ≥, you shade above the line. If the inequality symbol is < or ≤, you shade below the line.
To graph y > –2x + 1, start with the yintercept, which is 1. The slope is –2, which means –, so from the yintercept of 1, go down 2, because the slope is negative, and to the right 1. When you connect the starting point and the ending point, you will have the boundary line of your shaded area. You should extend this line as far as you'd like in either direction because it is endless. The boundary line will be dotted because the inequality symbol is >. If the symbol had been ≥, then the line would be solid, not dotted.
Checking Your Graph
When you graph, always check your graph to make certain the direction of the line is correct. If the slope is positive, the line should rise from left to right. If the slope is negative, the line should fall from left to right.
Now that you have the boundary line, will you shade above or below the line? The inequality symbol is >, so shade above the line.
If the inequality symbol is > or ≥, you will shade above the boundary line. If the inequality symbol is < or ≤, you will shade below the boundary line.
Special Cases of Inequalities
There are two special cases of inequalities. One has a vertical boundary line and the other has a horizontal boundary line. For example, the inequality x > 2 will have a vertical boundary line, and the inequality y > 2 will have a horizontal boundary line. The inequality y > 2 is the same as the inequality y > 0x + 2. It has a slope of 0 and a yintercept of 2. When the slope is 0, the boundary line will always be a horizontal line.
The inequality x > 2 cannot be written in y = form because it does not have a slope or a yintercept. It will always be a vertical line. It will be a dotted vertical line passing through the point on the xaxis where x = 2.
A horizontal line has a slope of 0. A vertical line has no slope.
Real World Inequality Graphs
When you graph an inequality, you have a picture of all possible answers to a problem. Can you see that it would be impossible to list them all? Think how valuable graphs are to the business world. If the shaded area on the graph represents all the possible prices a company can charge for a product and make a profit, executives can easily determine what to charge for a product to achieve the desired profit.
Find practice problems and solutions for these concepts at Functions and Graphs Practice Problems.