Understanding the Meaning of Absolute Value

One important concept in algebra that we learn is the distance between two points on the number line. In particular, we study the distance of a point that corresponds to a number to the point that corresponds to 0. In the following figure, the point (or circle) on the left represents 0, while the point on the right represents 5. To simplify our language, we will use coordinates to refer to its corresponding point on the number line. For instance, we will use -8 to refer to the point that corresponds to -8.

Looking at the number line, it is easy to see that 5 is 5 units away from 0 and that -8 is 8 units away from 0. We can also see that 0 is 0 units away from 0. The distance of a number from 0 on the number line is called its absolute value. Hence, the absolute value of 5 is equal to 5, the absolute value of -8 is equal to 8, and the absolute value of 0 is equal to 0. » Read more

Understanding Domain and Range Part 4

In this post, we summarize the previous three articles about domain and range. In the first part of the series, we focused on the graphical meaning of domain and range. We have learned that the domain of a function can be interpreted as the projection of its graph to the x-axis. Similarly, the range of the function is the projection of its graph to the y-axis.

Graphical meaning of domain (red) and range (green)

In the second part of the series, we learned to analyze equations of functions to determine their domain and range. We learned the restrictions in the domain and range of functions are affected by the following: squares in the expressions, square root signs, absolute value signs, and being in the denominator. In exploring these we concluded the following:

• Expressions under the square root sign result to a positive real number or 0.  This means that we have to set the inequality such that the expression is greater than or equal to 0, and then find the permissible values of x.
• Expressions containing squares result to a positive real number or 0. This affects the range of the function.
• Expressions inside the absolute value sign result to a positive number of 0. This also affects the range of the function.
• Expressions in the denominator of fractions cannot be 0 because it will make the function undefined. So, we need to find the value of x that makes the denominator by 0. To do this, we equate the expression in the denominator to 0 and find the value of x. The values of x are the restrictions in the domain.

In the third part of the series, we examined functions that have more complicated equations than those in the second part of the series.

Before I end this series, there is one more concept about domain that I want you to remember. That is, the domain of all polynomial functions is the set of real numbers. That’s why the domain of linear functions and quadratic functions in Part 1 and Part 2 is the set of real numbers.

Understanding Domain and Range Part 3

In the previous post, we have learned how to analyze equations of functions and determine their domain and range. We have observed that the range of the functions $y = x^2$ and $y = |x|$ are the set of real numbers greater than or equal to $0$ since squaring a number or getting its absolute value results to $0$ or a positive real number. We also learned that for a function to be defined,  the number under the square root sign must be greater than or equal to 0. Lastly, we have learned that we cannot divide by zero because it will make the function undefined.

In this post, we are going to continue our discussion by examining functions with equations more complicated than those in the second part of this series.

Squares and Absolute Values

1. $f(x) = x^2 - 3$

Domain: The function is defined for any real number $x$, so the domain of $f$ is the set of real numbers.

Range: The minimum value of $x^2$ is $0$ for any real number $x$ and $f(0) - 3 = 0^2 - 3 = -3$. So, the minimum value of the function is $-3$. We can make the value of the function as large as possible by increasing the absolute value of $x$. So, the range of the function is the set of real numbers greater than or equal to $-3$ or $[-3, \infty)$ in interval notation.  » Read more

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