## The Basics of Direct Proportion

A car travels at 40 kilometers per hour traveled for 7 hours. The table of the distances traveled with respect to time is shown below.

Observe the relationship between the distance traveled and the time. As the time increases, the distance traveled increases. If the number of hours increases two times, then the distance also increases two times. Between hour 1 and hour 2, the increase in time is 1 hour, and the increase in distance is 40 kilometers. Between hour 3 and hour 5, the increase in time is 2 hours, and the increase in distance is 2(40) = 80 kilometers.

If there are two changing quantities $x$ and $y$ and if the value of $x$ changes 2 times, 3 times, and so on, $y$ also changes 2 times, 3 times, and so on respectively, we can say that $y$ is directly proportional to $x$. In the relationship above, distance is directly proportional to time.  » Read more

## Proof Without Words: If x >0, x + 1/x >=2

A picture is worth a thousand words, even in mathematics. Sometimes, complicated algebraic expressions can be shown using very simple geometric representations. One example is the sum

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$

which equals 1.

In this post, we show another proof without words. Proof without words is not a proof per se but rather a representation that will help readers understand a proof. » Read more

## What is the horizontal line test?

In January of this year, we have discussed about the vertical line test. We have learned that if a vertical line intersect a graph more than once, then that graph is not a function. In this post, we learn about the horizontal line test and its relation to inverse functions.

Suppose we have a function $f$. Then, we input $x$ and call the output $f(x)$. If we do things backward, suppose we have the number $y$ which is in the range of $f$. What value should we input to $f$ to get $y$? Let’s have a more specific example.

Suppose we have the function $f(x) = x^2$. And we choose the number $y = 9$ which is in the range of $f$. What number should we input in $f$ to get $9$? Well, we will have two numbers, those are $3$ and $-3$. » Read more