Mathematics: Levels E, M, D, and A Study Guide for the TABE Test
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Functions
Percentage of Test Level Specifically Assessing Functions (— = Assumed)
| L | E | M | D | A |
|---|---|---|---|---|
| not tested | not tested | not tested | 11% | 28% |
A function has three elements: the input, the output, and their relationship. Every input relates to an output. They are also called ordered pairs and can be represented by \((x, f(x))\). 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, such as:
\(g(r) = r + 5\)
\(h(\theta) = \theta^2\)
A nice simple way to think of a function is to think of it as a device that takes a number that is fed into it (the input) and does something with the number to produce a new number (the output).
When a function is written, it 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:
\(f(x) = x-2\) tells you that if you input a number for \(x\) the function will output the value of \(x-2\). If you decide to input the number \(6\), this is how the problem would be written:
\[f(6) = 6-2\] \[f(6) = 4\]In this function, “\(f(n) = - 3n + 10\)”, what is \(f(n)\) if \(n = -7\)?
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 \(31\).
Basic Function Ideas
We have seen the equation \(y=mx+b\) for a straight line before. This equation can also be written as a function of \(x\): \(f(x) = mx+b\). Often the words equation and function can be used interchangeably.
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. That’s how we can know that the equation \(y=x^2+3x-4\) is not linear. In fact, if you graph it, you will get a parabola.
Constructing Functions
Suppose a bicyclist rides by and you are able to measure how far she goes in certain times. You record your results in a table and make a graph of the data.
| Distance (ft) | Time (s) |
|---|---|
| 0 | 0 |
| 14 | 1 |
| 28 | 2 |
| 42 | 3 |
| 70 | 5 |
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 \(14\) ft in \(1\) s. That’s a ratio of:
\[\frac{14 \text{ ft}}{1 \text{ s}} = 14 \frac{\text{ ft}}{\text{ s}}\]We can do the same for any value in the table; for example, the last pair of values will give us:
\[\frac{70 \text{ ft}}{5 \text{ s}} = 14 \frac{\text{ ft}}{\text{ s}}\]It’s worth reminding ourselves of a few other things. First, the rate of change in this example is also the slope of the graph (rise/run). Second, this rate of change has a physical meaning: \(\frac{\text{ ft}}{s}\) is a unit of speed. Third, the fact that the rate of change is the same for any of the pairs of values means that the bike was going at a constant speed. That’s also why the graph is a straight line.
Describing Functions
Given a function statement, you should be able to describe the general relationship between two quantities. For example, instead of the function being \(f(t) = 14 t\), 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 is nonlinear, getting steeper as time goes by? That means she is picking up speed as she rides. (Remember, the slope tells us the speed.)
For a given function you should be able to tell whether it is increasing or decreasing from left to right and whether it is linear or nonlinear.
Domain and Range
As mentioned before, 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 table of values for the function shown.
\[f(x) = 2x +3\] \[\begin {array}{|c|c|} \hline \text{Domain}&\text{Range}\\ \hline x & f(x)\\ \hline 1 & 5\\ \hline 2 & 7\\ \hline 3 & 9\\ \hline 4 & 11\\ \hline \end{array}\]There’s a technicality here that you need to be aware of. There are relationships that are not functions. You could use them to make a data table that would look a lot like the ones you’ve already seen for functions. But the table below does not show a function.
Suppose some relationship between \(x\) and \(y\) gives these numbers:
\[\begin {array}{|c|c|} \hline \text{Domain}&\text{Range}\\ \hline x & f(x)\\ \hline 6 & 3\\ \hline 5 & 8\\ \hline 6 & 5\\ \hline 8 & 12\\ \hline \end{array}\]What makes it not a function? This: 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 Notation
You’ve seen function notation a few times. Customarily, the \(f(x)\) part is written on the left, and on the right there is an expression showing what to do with the input.
Here is a function notation and its translation:
\[\begin{array}{rl} f(x) = & x^2 -5 \\ \text{There is a function of } x \text{ that tells us to} \text{ square } x \text{ and subtract } 5\\ \end {array}\]You should be able to find the value of a function given an input value. (Remember, any letter can be used in the function, not always \(x\)). 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:
\[f(5) = 2(5)^2 + 3 \cdot 5 +7\] \[f(5) = 2 \cdot 25 +15+7\] \[f(5) = 50+15+7\] \[f(5) = 72\]Function Displays
We’ve seen examples of function displays in this guide. We’ve seen function notation, an example being \(f(x) = x^2 +7\). We’ve also seen data tables and graphs. This section provides another look at them, summarizing what we’ve seen and adding a few details.
Graphs
Given a graph of a real event, 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 after 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.
- The rock is not moving: Point A, because at that exact point the slope of the graph is zero.
- The rock is slowing down: Point B, because the slope is getting less steep as the graph rises.
- The rock is speeding up: Point D, because the slope is getting steeper as the graph goes down.
Also:
How long was the rock above the astronaut’s hand? \(10\) s.
- How long was the rock moving upward? \(5\) s.
- What was the rock’s maximum height? Approximately \(124\) ft.
- How would you describe the path of the rock? A vertical line up and down. (The graph isn’t the path of the rock, it’s only showing how the height changes with time.)
Given a description of an event, can you draw a graph to represent it?
Event: 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. You should be able to make a distance-time graph for this event.
The graph should look a lot like this one. The car was going at a steady speed, so the line should be straight. It should go upward because the distance is increasing as time goes by. The timer started at a distance of zero and a time of zero. The graph ends at the point (5, 200).
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 | 34.7 |
What could you say 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 to calculate the slopes of the first three data entries.
\[\frac{20-0}{8.7-0} =\frac{20}{8.7} = 2.31\] \[\frac{40-20}{17.4-8.7} = \frac{20}{8.7} = 2.31\] \[\frac{60-40}{26-17.4} = \frac{20}{8.7} = 2.31\]If you tried the fourth pair, you would get the same result, so, yes, it is a linear relationship.
What about the air pressure on us? It gets less as we increase our altitude. How would a graph of altitude vs. air pressure look compared to a graph of depth vs. 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 this would take a table like this one.
| Altitude (ft) | Air Pressure (psi) |
|---|---|
| 0 | 14.7 |
| 10,000 | 10.1 |
| 20,000 | 6.8 |
| 30,000 | 4.4 |
| 40,000 | 2.7 |
We’ll try the ratio thing again, calculating the change in air pressure compared to the change in altitude.
\[\frac{10.1-14.7} {10\text{,}000-0}= \frac{4.6}{10\text{,}000} = -0.00046 \frac{\text{psi}}{ft}\] \[\frac{ 6.8-10.1}{20\text{,}000-10\text{,}000} = \frac{3.3}{10\text{,}000}{3.3} = -0.0003.3 \frac{\text{psi}}{ft}\] \[\frac{4.4-6.8}{30\text{,}000-20\text{,}000} = \frac{2.4}{10\text{,}000} = -0.00024\frac{\text{psi}}{ft}\] \[\frac{2.7 - 4.4}{40\text{,}000-30\text{,}000} = \frac{1.7}{10\text{,}000} = -0.00017\frac{\text{psi}}{ft}\]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 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. (Ignore the negative signs, a negative slope just tells us that the graph is going downhill.)
Rate of Change
We’ve seen rates of change a few times already in this guide. To summarize, to find a rate of change, you first pick two points on a graph or in a data table and use them in this way:
\[\frac{\text{Change in } y \text{ or }{ f(x)}} {\text{Change in }x}\]Or, in terms of columns in a table:
\[\frac{\text{Change in right column}} {\text{Change in left column}}\]Or, in terms of a graph:
\[\frac{\text{Change in vertical axis}} {\text{Change in horizontal axis}}\]| Altitude (ft) | Air Pressure (psi) |
|---|---|
| 0 | 14.7 |
| 10,000 | 10.1 |
| 20,000 | 6.8 |
| 30,000 | 4.4 |
| 40,000 | 2.7 |
\[\frac{6.8-10.1}{20\text{,}000 - 10\text{,}000} = \frac{-3.3}{10\text{,}000} = -0.0003 \frac{\text{ psi}}{\text{ ft}}\]What is the rate of change in the table above between \(10\text{,}000\) ft and \(20\text{,}000\) ft.
Properties
Here we take a look at a few properties of linear and exponential functions and show interpreting the parameters in those functions.
Properties of Exponents and Functions
An exponential function has a variable for an exponent. \(f(x) = 4 \cdot 5^x\) is an exponential function.
On the other hand, \(y= 3x^4\) is not an exponential function. The \(x\) is not an exponent.
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. Here’s an example of exponential growth:
\[y= 2 \cdot 3^x\] \[\begin{array}{c|c|c|c|c|c|} &x&0&1&2&3&4\\ \hline &y&2 \times3^0&2 \times 3^1&2 \times 3^2&2 \times 3^3&2 \times 3^4\\ \hline &y&2&6&18&54&162\\ \end{array}\]If \(a=2\) and \(b=3\), then \(y= 2 \cdot 3^x\). The function would produce this data table:
Each \(y\) value is \(3\) times the \(y\) before it.
Here is an example of an exponential decay.
Suppose there is a sample of radioactive element Z that has a mass of \(10\) g, and every hour half of it disappears. This table shows data for its disappearance.
The equation for this follows the same pattern as \(y = ab^x\), but \(m\), for mass, will take the place of \(y\), \(a\) will be \(10\), \(b\) will be \(\frac{1}{2}\), and \(t\), for time, will take the place of \(x\).
\[m=10 \cdot (\frac{1}{2})^t\] \[\begin{array}{c|c|c|c|c|c|c|} &t&0&1&2&3&14&5\\ \hline &m&10\times(\frac{1}{2})^0&10\times(\frac{1}{2})^1&10\times(\frac{1}{2})^2&10\times(\frac{1}{2})^3&10\times(\frac{1}{2})^4&10\times(\frac{1}{2})^5\\ \hline &m&10&5&2.5&1.25&0.625&?\\ \end{array}\]You can see how the mass of Z keeps decreasing. That’s because \(b\) is less than \(1\)
Properties of Two Functions
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?
\(2y=6x-4\) or \(3x-8y = 10\)?
Put them both in \(y=mx+b\) form and compare the \(m\) values.
First equation: Divide through by \(2\) and get \(y=3x-2\). Here, \(m=3\).
Second equation: Rearrange to get \(-8y=-3x+10\). Divide through by \(-8\) and get \(y=\frac{3}{8} m -\frac{5}{4}\). Here \(m=\frac{3}{8}\). The first equation produced a steeper slope.
Let’s try a different comparison.
We have two different functions, A and B:
Function A: \(y=2x-7\)
Function B: The function that makes this data table:
Which function has the larger slope? Pick two points in the data table, say \((1, 4)\) and \((2,7)\), and use them to find the slope for that function.
\[m=\frac{(y_2-y_1)}{(x_2-x_1)} = \frac{(7-4)}{(2-1)} = 3\]The second function has a larger slope.
Growth and Decay
There are situations in which quantities grow or shrink by a constant percentage. For growth, this means that each step is always the same percentage higher than the step before. For a classic example, let’s take a look at money, specifically what is called compound interest.
Here’s how it works: Say you put \(\$1\text{,}000\) in the bank and the bank will pay you \(2\%\) interest each year. Each year the bank will pay you \(2\%\) of the amount in your account. It will look like this (the numbers do get a little messy, just notice that each year adds \(0.02\%\) to the total):
\[\begin {array}{|c|c|c|} \hline \text{At the end of year} & \text{You get this interest} & \text{Now you have}\\ \hline 1 & .02 \times \$1\text{,}000.00 = \$20.00 & \$1\text{,}020.00\\ \hline 2 & .02 \times \$1\text{,}020.00 = \$20.40 & \$1\text{,}040.40\\ \hline 3 & .02 \times \$1\text{,}040.40 = \$20.81 & \$1\text{,}061.21\\ \hline 4 & .02 \times \$1\text{,}061.21 = \$21.22 & \$1\text{,}082.24\\ \hline \end{array}\]You’re probably not going to get rich this way.
You can see a constant percentage decay in the sequence below. Each number is \(5\%\) less than the one before.
\[100,\ 95,\ 90.25,\ 85.74\]
Five percent of \(100 = 5 \text{, and } 100 - 5 = 95\)
Five percent of \(95 = 4.75 \text{, and } 95 - 4.75 = 90.25\)
Five percent of \(90.25 = 4.51 \text{, and } 90.25- 4.51 = 85.74\)
Function Parameters
Be able to interpret parameters in functions, both linear and exponential, when they apply to an actual situation.
Parameters are the constants in an equation or function that define the behavior of the function. By constants, we mean the given numerical values in the function. In \(y=3x-2\), \(3\) and \(-2\) are the constants. Very often, in a function, constants are indicated by using letters from the beginning of the alphabet: a, b, c, d, etc. An exception is the use of \(m\) for slope in a linear function.
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.
For a graph, \(m\) 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 \(f(x)\) at the beginning, when \(x=0\).
So, let’s interpret those parameters in this situation.
\[y = -300x +700\]Suppose 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:
Which parameter tells us the speed of the airplane? Remember, in \(y=mx + b\), \(m\) is the rate of change, which, in this case, is the speed: \(-300\). (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 (\(x =\) zero hours), there are \(700\) miles to go.
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 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 \(\$10\), a function that will work is \(y= 10(1.04)^x\), 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.
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 \(y\):
\[y=10(1.04)^5\] \[y = 10(1.22)\] \[y= 12.20\]All Study Guides for the TABE Test are now available as downloadable PDFs