Mathematics Study Guide for the TABE Test

Algebraic Concepts: Functions

In algebra, functions are similar to equations. In fact, often the words are used interchangeably. However, instead of an expression being equal to a value or another expression, in a function we are looking for the result of an action, the input.

Function Notation

A function has three elements: the input, the output, and their relationship. Every input relates to one specific output. The usual way of representing a function is to write it as \(f(x)\), read as “\(f\) of \(x\)”, but it can be written in many other ways. Functions are also called ordered pairs and can be represented by \((x, f(x))\). These are both examples of functions:

\[g(r) = r + 5\] \[h(\theta) = \theta^2\]

Think of a function as a machine that takes a number that is fed into it (the input) and produces a new number (the output). A written function gives an expression that tells you exactly what to do with the input to produce the output. All you need to do is take the input number and substitute it into the given expression. For example:

\[f(x) = x-2\]

This tells you that if you input a number for \(x\), the function will output the value of \(x-2\). If you input the number \(6\), this is the result:

\[f(6) = 6-2\] \[f(6) = 4\]

You should be able to find the value of a function given an input value. That kind of problem will look like this:

If \(f(t) = 2t^2 +3t +7\), evaluate \(f(5)\).

All we do is substitute \(5\) for \(t\) in the expression on the right:

\[f(5) = 2(5)^2 + 3 \cdot 5 +7\] \[f(5) = 2 \cdot 25 +15+7\] \[f(5) = 50+15+7\] \[f(5) = 72\]

Remember, as always, to follow the order of operations (PEMDAS).

Let’s try another example.

In the function \(f(n) = - 3n + 10\), what is \(f(n)\) if \(n = -7\)?

Solution

Substitute the value of \(n\) in the equation to get \(f(n)\):

\[f(-7) = -3\cdot(-7) + 10 = 31\]

This means that an input of \(-7\) for this function produces an output of \(31\).

Constructing Functions

In truth, a function is closely related to something you’ve seen before, independent and dependent variables. As such, you should already know how a function will look when put in a chart or graph.

Suppose a bicyclist rides by and you are able to measure how far she goes in certain times. You record your results in the following table and make a graph of the data:

Distance (ft)	Time (s)
\(0\)	\(0\)
\(14\)	\(1\)
\(28\)	\(2\)
\(42\)	\(3\)
\(70\)	\(5\)

You’ve seen such a graph before. It looks just like the graph of a linear equation. Now, though, we can also construct a function to model the linear relationship between distance and time. Looking at the data table, notice that the distance is the time multiplied by \(14\). We can write \(d = 14t\). In function notation that would be:

\[f(t) = 14t\]

The rate of change is the ratio of the change in distance to the change in time. If we look at the second value in the data table, we see the bicyclist traveled \(14\) feet in one second. That’s a ratio of:

\[\frac{14 \text{ ft}}{1 \text{ s}} = 14 \text{ ft}/\text{ s}\]

As you can see, ratios and functions are closely related. Just as this ratio tells you how distance and time are related, a function tells you how the input (time) is related to the output (distance).

Linear Equations as Functions

We have seen the equation \(y=mx+b\) for a straight line before. This equation can also be written as:

\[f(x) = mx+b\]

Any linear equation can be considered a function. The linear equation \(y = 3x-6\) can be written as the linear function \(f(x) = 3x-6\). As long as the \(x\) has a power of \(1\), the function will be linear.

Describing Functions

Given a function statement, you should be able to describe the general relationship between two quantities. You should be able to tell whether it is increasing or decreasing from left to right and whether it is linear or nonlinear (not a straight line).

For example, instead of the function being \(f(t) = 14 t\) (a linear function), let’s say the bicyclist above is following the function \(f(t) = 5t^2\). We’ll graph it to help see what it shows. See how the graph curves, getting steeper as time goes by? That means it is nonlinear. What this tells us in real-world terms is that the bicyclist is picking up speed as she rides. In other words, when the graph is getting steeper, we can see that the distance traveled per second is increasing.

Domain and Range

As discussed, functions have inputs and outputs. The set of inputs that can be chosen is called the domain of the function. The set of outputs is called the range of the function. Here is a function and the table of values for that function:

\[f(x) = 2x +3\] \[\begin {array}{|c|c|} \hline \text{Domain}&\text{Range}\\ \hline x & f(x)\\ \hline 1 & 5\\ \hline 2 & 7\\ \hline 3 & 9\\ \hline 4 & 11\\ \hline \end{array}\]

There’s a technicality here that you need to be aware of. There can be relationships that are not functions. You could use such relationships to make a data table that would look a lot like the ones you’ve already seen for functions. Suppose some relationship between \(x\) and \(y\) gives these numbers:

\[\begin {array}{|c|c|} \hline \text{Domain}&\text{Range}\\ \hline x & f(x)\\ \hline 6 & 3\\ \hline 5 & 8\\ \hline 6 & 5\\ \hline 8 & 12\\ \hline \end{array}\]

This might look like a function, but it is not. Why? Because one input gives two different outputs. The input \(6\) gives \(3\) for an output, but it also gives \(5\) for an output. For a function, one input always gives only one output. If you see a data table with the same number twice on the input side (domain), that table doesn’t represent a function.

Function Displays

As you’ve seen, functions can be graphed in various ways. Being able to interpret a graph of a function is an important skill and one that you’ll want to develop before taking the TABE.

Graphs

Given a graph of a real event, you will need to be able to describe what is physically happening at different points on the graph. The graph below represents a moon rock thrown straight up by an astronaut on the Moon who then caught it when it came back down. The timer started at the instant the rock was leaving the astronaut’s hand, and it ended at the instant it was caught:

In the graph above, can you tell which letter corresponds to each part of the event? Remember, in a distance-time graph, slope equals speed. Here is how to read the graph:

• Points A and E: The rock is zero feet above the astronaut since these points are on the \(x\)-axis (and so is the astronaut). Therefore, the rock is not moving because it is in the hands of the astronaut.

• Point C: At this point, the rock is at the apex of its path of travel, neither going up nor going down.

• Point B: The rock is slowing down, because the curve is getting less steep as the graph rises, indicating that the height increase per second is slowing down.

• Point D: The rock is speeding up, because the curve is getting steeper as the graph goes down, indicating that the height decrease per second is speeding up.

The graph tells us even more information. For instance, we know how long the rock was out of the astronaut’s hand: \(10\) seconds. Other data points we can learn from this graph include:

• How long the rock was moving upward: five seconds.
• The rock’s maximum height: approximately \(124\) feet.

Now that you know how to read a graph, let’s see if you can take a description of an event and graph it.

You are standing beside a road and a car goes by at a steady speed. You start a timer at that instant. Five seconds later you stop the timer when the car has gone \(200\) feet. Create a distance-time graph for this event.

Solution

We’re creating a distance-time graph, so let’s make the vertical axis “Distance (ft)” and the horizontal axis “Time (s).” You’re told the car was going at a steady speed (distance divided by time), so right away you know the line to be graphed will be straight. It should go upward because the distance is increasing as time goes by. The timer started at a distance of \(0\) and a time of \(0\). The graph needs to end at the point \((5, 200)\). So, our graph will look like this:

Tables

The skills mentioned above also apply to data tables. You should be able to look at a data table and describe the relationship between the two quantities shown. For example, scuba divers are very aware that as they go deeper in water, the pressure on them increases. This table describes the relationship between depth and pressure (psi is the abbreviation for pounds per square inch):

Depth (ft)	Pressure (psi)
\(0\)	\(0\)
\(20\)	\(8.7\)
\(40\)	\(17.4\)
\(60\)	\(26.1\)
\(80\)	\(34.8\)

What can you determine about the relationship? Well, clearly, as depth increases, so does pressure. Is it a linear relationship? We could graph it and see, or we could see if the depth-to-pressure ratio stays the same. If it does, then the relationship is linear. Let’s try a few and see. We will use the slope formula (\(m= \frac{y_2-y_1}{x_2-x_1}\)) to calculate the slopes of the first three data entries:

\[\frac{8.7-0}{20-0} =\frac{8.7}{20} = 0.435 \frac{\text{psi}}{\text{ft}}\] \[\frac{17.4-8.7}{40-20} =\frac{8.7}{20} = 0.435 \frac{\text{psi}}{\text{ft}}\] \[\frac{26.1-17.4}{60-40} =\frac{8.7}{20} = 0.435 \frac{\text{psi}}{\text{ft}}\]

If you tried the fourth pair, you would get the same result, so, yes, it is a linear relationship.

What about air pressure? When you go up in a plane, it reduces as your altitude increases, as you can see in this table:

Altitude (ft)	Air Pressure (psi)
\(0\)	\(14.7\)
\(10\text{,}000\)	\(10.1\)
\(20\text{,}000\)	\(6.8\)
\(30\text{,}000\)	\(4.4\)
\(40\text{,}000\)	\(2.7\)

How would a graph of altitude versus air pressure look compared to a graph of depth versus water pressure? The big difference is that as altitude goes up, air pressure goes down, so this graph would go downhill. Is it linear? To answer that question, let’s use the table info and the ratio trick we used before:

\[\frac{10.1-14.7} {10\text{,}000-0}= \frac{-4.6}{10\text{,}000} = -0.00046 \frac{\text{psi}}{\text{ft}}\] \[\frac{ 6.8-10.1}{20\text{,}000-10\text{,}000} = \frac{-3.3}{10\text{,}000} = -0.00033 \frac{\text{psi}}{\text{ft}}\] \[\frac{4.4-6.8}{30\text{,}000-20\text{,}000} = \frac{-2.4}{10\text{,}000} = -0.00024\frac{\text{psi}}{\text{ft}}\] \[\frac{2.7 - 4.4}{40\text{,}000-30\text{,}000} = \frac{-1.7}{10\text{,}000} = -0.00017\frac{\text{psi}}{\text{ft}}\]

We can see that the ratios are nowhere near constant, so this is not a linear relationship. Did you notice that we were calculating the slope of each section of the data table? What does the negative sign mean in the slope? The graph is going downhill.

We also know the graph isn’t straight, but which way does it curve? Steeper and steeper, or flatter and flatter? Flatter and flatter, because the slopes are decreasing.

Rate of Change

We’ve discussed rate of change briefly earlier. This concept plays an important role when it comes to functions, and it is something that we can see clearly when using either graphs or tables. To summarize, to find a rate of change, you divide the change in \(y\) or \(f(x)\) by the change in \(x\).

In terms of a table, you divide the change in the right column by the change in the left column. In terms of a graph, you divide the change in the vertical distance (\(y\)-value) by the change in the horizontal distance (\(x\)-value). Let’s try a sample problem.

What is the rate of change in the table between \(10\text{,}000\) feet and \(20\text{,}000\) feet.

Altitude (ft)	Air Pressure (psi)
\(0\)	\(14.7\)
\(10\text{,}000\)	\(10.1\)
\(20\text{,}000\)	\(6.8\)
\(30\text{,}000\)	\(4.4\)
\(40\text{,}000\)	\(2.7\)

Solution

Simply divide the rate of change of the right column by the rate of change in the left column:

\[\frac{6.8-10.1}{20\text{,}000 - 10\text{,}000} = \frac{-3.3}{10\text{,}000} = -0.0003 \frac{\text{psi}}{\text{ft}}\]

Properties of Functions

Here we take a look at a few properties of different types of functions and show how to interpret the parameters in those functions.

Exponential Functions

An exponential function has a variable for an exponent. For instance, \(f(x) = 4 \cdot 5^x\) is an exponential function, as the exponent is \(x\). On the other hand, \(y= 3x^4\) is not an exponential function, as the exponent is a number.

An exponential function with \(x\) and \(y\) takes the form \(y = ab^x\), where \(a\) and \(b\) are constants. If \(b\) is greater than \(1\), the value of \(y\) will increase as \(x\) increases, and the function is called an exponential growth function. If \(b\) is less than \(1\), the value of \(y\) will decrease as \(x\) increases, and the function is called an exponential decay function. The following scenario illustrates exponential growth:

If \(a=2\) and \(b=3\), then \(y= 2 \cdot 3^x\). The function would produce this data table:

\[y= 2 \cdot 3^x\] \[\begin{array}{c|c|c|c|c|c|} &x&0&1&2&3&4\\ \hline &y&2 \times3^0&2 \times 3^1&2 \times 3^2&2 \times 3^3&2 \times 3^4\\ \hline &y&2&6&18&54&162\\ \end{array}\]

Each \(y\) value is three times the \(y\) before it.

We can also see exponential decay in a real-world situation.

Suppose there is a sample of radioactive element Z that has a mass of \(10\) grams, and every hour half of the element disappears. An equation to describe this situation will follow the same pattern as \(y = ab^x\), but \(m\), for mass, will takes the place of \(y\), and \(t\), for time, will takes the place of \(x\). We insert \(10\) for \(a\) and \(\frac{1}{2}\) for \(b\) and get:

\[m=10 \cdot (\frac{1}{2})^t\]

This table shows data for the element’s disappearance:

\(t\)	\(m\)
\(0\)	\(10 \times \left(\frac{1}{2}\right)^0 = 10\)
\(1\)	\(10 \times \left(\frac{1}{2}\right)^1 = 5\)
\(2\)	\(10 \times \left(\frac{1}{2}\right)^2 = 2.5\)
\(3\)	\(10 \times \left(\frac{1}{2}\right)^3 = 1.25\)
\(4\)	\(10 \times \left(\frac{1}{2}\right)^4 = 0.625\)
\(5\)	\(10 \times \left(\frac{1}{2}\right)^5 = 0.3125\)

You can see how the mass of Z keeps decreasing. That’s because \(b\) is less than \(1\).

Comparing Two Functions

You will also need to be able to compare two different functions that are in different forms. They may be represented by algebraic equations, graphs, data tables, or even a verbal description.

To compare functions, it’s a good idea to have them in the same form. For example, which of the following equations would have a steeper graph?

\[2y=6x-4\] \[3x-8y = 10\]

The easiest way to compare them would be to put them both in the slope-intercept form (\(y=mx+b\)) and compare the \(m\) values.

To do that, let’s divide the first equation through by \(2\), which gives us:

\[y=3x-2\]

Here, \(m=3\).

For the second equation, rearrange it to get \(-8y=-3x+10\). Now, divide through by \(-8\) and get:

\[y=\frac{3}{8} m -\frac{5}{4}\]

Here, \(m=\frac{3}{8}\). Therefore, the first equation produced a steeper slope.

Let’s try another comparison.

Which of these two functions, A and B, has a steeper slope:

Function A: \(y=2x-7\)
Function B: The function makes this data table:

\[\begin {array}{|c|c|c|c|c|} \hline x&0&1&2&3\\ \hline y&1&4&7&10\\ \hline \end{array}\]

Solution

Function A is already in the slope-intercept form, so we know its slope is \(2\). For Function B, let’s pick two points in the data table, say \((1, 4)\) and \((2,7)\), and use them to find the slope:

\[m=\frac{(y_2-y_1)}{(x_2-x_1)} = \frac{(7-4)}{(2-1)} = 3\]

Therefore, Function B has a steeper slope.

Function Parameters

The parameters of a function (or equation) are the constants and coefficients that define the behavior of the function. In \(y=3x-2\), \(3\) and \(-2\) are the parameters (\(3\) is a coefficient and \(2\) is a constant). Very often, in a function, parameters are indicated by using letters from the beginning of the alphabet. An exception is the use of \(m\) for slope in a linear function.

Parameters of Linear Functions

Speaking of linear functions, we have seen a few times that the general equation of a linear function is \(y = mx + b\). The parameters here are \(m\) and \(b\). The \(m\) tells us the rate of change of \(y\), and the \(b\) tells us the value of \(y\) when \(x\) is zero.

When graphing a function, \(m\) is the slope of the graph and \(b\) is the \(y\)-intercept. The parameters \(m\) and \(b\) tell us exactly where the graph of \(f(x)\) lies. If \(m\) is positive, \(y\) is increasing. If \(m\) is negative, \(y\) is decreasing. The parameter \(b\) tells us the value of \(f(x)\) when \(x=0\).

Consider this situation:

You are flying to Seattle from a place that is \(y\) miles away, and \(x\) is the number of hours spent in the air. The longer you fly, the fewer miles there are to go. This equation describes the trip:

\[y = -300x +700\]

Which parameter tells us the speed of the airplane? Remember, in \(y=mx + b\), \(m\) is the rate of change (or speed), which, in this case, is \(-300\). The negative just means that the distance is decreasing.

What is happening when \(x=0\)? When \(x=0\), the equation is \(y=-300(0) +700\), which is just \(y=700\). That tells us that at the beginning of the trip (\(x = 0\) hours), there are \(700\) miles to go.

Parameters of Exponential Functions

In an exponential function, the general function is \(y=ab^x\). The parameters here are \(a\) and \(b\). The \(b\) parameter is the growth or decay factor:

• If \(b\) is greater than \(1\), it is a growth factor and \(y\) will increase.
• If \(b\) is between \(0\) and \(1\), it is a decay factor and \(y\) will decrease.

What happens to \(y\) when \(x=0\)?

\[y=ab^0\] \[y= a(1)\] \[y = a\]

So, \(a\) is the value of \(y\) when \(x=0\).

Now we will look at a situation that uses an exponential function. As you may know, prices often rise every year, so let’s write a function to figure out what a price will be some years down the road if the price increases by \(4\%\) each year. The parameter \(b\) will be a growth factor, so it will be greater than \(1\) in our function.

If the starting price, parameter \(a\), is \(\$10\), we can use this function:

\[y= 10(1.04)^x\]

where \(y\) is the new price and \(x\) is the number of years down the road. The \(1.04\) comes from the \(4\%\) increase each year (\(1+0.04\)).

What price are we starting with? At the start, \(x=0\), so put \(0\) in our function for \(x\):

\[y=10(1.04)^0\] \[y=10(1)\] \[y=10\]

If our starting price is \(\$10\), how much will it be in five years? Put \(5\) in for \(x\) and find \(y\):

\[y=10(1.04)^5\] \[y = 10(1.22)\] \[y= 12.20\]