Math and Multimedia

Understanding Domain and Range Part 3

August 7, 2016 GB Calculus and Analysis, College Algebra, High School Algebra, High School Calculus, High School Mathematics

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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Understanding Domain and Range Part 2

July 31, 2016 GB Calculus and Analysis, High School Algebra, High School Calculus, High School Mathematics

In the previous post, we have learned the graphical representation of domain and range. The domain of the function $f$ is the shadow or projection of the graph of $f$ to the x-axis (see the red segment in the figure below). The range of $f$ is the projection of the graph of $f$ to the y-axis (see the green segment in the figure below). In this post, we are going to learn how to analyze equations of functions and determine their domain and range without graphing.

domain and range

If a graph of a function is projected to the x-axis, the projection is the set of x-coordinates of the graph. A single point $(a,0)$ on the projection means a point on the graph exists. The existence of a point implies that $f(a)$ exists. This means that the function is defined at $x = a$ . In effect, the domain of a function is the set of x-coordinates that makes the function defined. In what follows, we learn some examples to illustrate this concept. » Read more

Leave a comment domain of a function examples, domain of a square root function, domain of functions, domain of rational function, getting the domain of a function, range of a function examples, range of functions

Understanding Domain and Range Part 1

July 23, 2016 GB College Algebra, High School Calculus, High School Mathematics

The domain of a function is the set of x-coordinates of the points in the function. The range of the function f is the set of y-coordinates of the points in the function. So if we have a function f with points (-3, -2), (-1, 3), (2, 3), and (5,4), then the domain of the function f is the set {-3, -1, 2, 5} and the range of f is the set {-2, 3, 4). Graphically, we can say that the domain is the “projection” of the points to the x-axis (see red points in the following figure).

The range of f is the projection of the points to the y-axis (see green points in the following figure). » Read more

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