How to Find the Range of an Absolute Value Function
How to Find the Range of an Absolute Value Function
The absolute value function is a fundamental concept in mathematics that measures the distance of a number from zero on the number line. In this article, we will explore how to find the range of an absolute value function and uncover its applications in various fields.
Understanding Absolute Value Functions
Before diving into the range, let’s understand what an absolute value function is. An absolute value function can be expressed in the form:
f(x) = |ax + b| + c
where a, b, and c are constants. The function essentially outputs the distance of ax + b from zero, shifted vertically by c.
Steps to Find the Range of an Absolute Value Function
Step 1: Identify the Vertex
The vertex of the function gives the minimum or maximum value of the absolute value expression. For the function f(x) = |ax + b| + c, the vertex occurs at:
x = -b/a
By substituting this value back into the function, you can find the y-coordinate of the vertex:
f(-b/a) = c
Step 2: Determine the Minimum Value
For absolute value functions, the minimum value is determined by the c term when a is greater than zero. Thus, the minimum value of f(x) is:
Minimum = c
Step 3: Analyze Behavior as x Approaches Infinity
As x approaches positive or negative infinity, the value of f(x) will also approach infinity. Therefore, the function has no upper limit.
Conclusion: The Range of Absolute Value Functions
By following these steps, we find that the range of an absolute value function is:
Range: [c, ∞)
This means the function outputs values starting from c and extends to positive infinity.
Practical Example
Consider the absolute value function:
f(x) = |2x + 3| - 5
The vertex occurs at x = -3/2, and substituting gives:
f(-3/2) = -5
Hence, the minimum value is -5 and the range of this function is:
Range: [-5, ∞)
Understanding how to find the range of an absolute value function is essential for students and professionals alike. Hopefully, this article clarified the concept and provided a clear method for determining the range.
Further Reading
If you want to learn more about absolute value functions and their properties, here are some additional resources: