Absolute Value Functions

An Insight Into Absolute Value Functions

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our galaxy, some tens of thousands of light-years away. Then, the astronomer Edwin Hubble proved that these objects are actually galaxies in their own right, at distances of millions of light-years. Today, the situation is that astronomers can easily detect galaxies that are billions of light-years away. Distances in the universe can be measured in all directions. It is useful to consider distance as an absolute value function.

Introduction: Absolute Value Functions

Suppose you want to have a proper understanding of absolute value functions. First, you need to know its features. The absolute value function is one of the toolkit functions. The absolute value function is commonly thought to provide the distance with the number from zero on a number line. If you consider it algebraically, the output is the value without regard to the sign for whatever the input value is.

The general form of an absolute value function is f(x)=a|x-h|+k. From this form, we can easily draw graphs.

The absolute value function serves to make any inserted value a positive value. So, you can go for the review and analyze the meaning of absolute value in mathematics, learn how to construct, calculate, graph, and absolute value function, and discover the four characteristics of all absolute value functions.

To solve an equation containing the absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both, calculate the positive and negative value of the number on the other side of the equation, and solve both equations.

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Graphs Of Absolute Value Functions

The graph of the absolute value function f (x) = | x| is similar to the graph of f (x) = x, except that the "negative" half of the graph is reflected over the x-axis. Here is the graph of f (x) = | x|:

f (x) = x

The graph looks like a "V", with its vertex at (0, 0). Its slope is m = 1 on the right side of the vertex and m = - 1 on the left side of the vertex.

We can translate, stretch, shrink, and reflect the graph.

Here is the graph of f (x) = 2| x - 1| - 4:

f (x) = 2| x - 1| - 4 F

Here is the graph of f (x) = - | x + 2| + 3:

f (x) = - | x + 2| + 3:

f (x) = - | x + 2| + 3

In general, the graph of the absolute value function f (x) = a| x - h| + k is a "V" with vertex (h, k), slope m = a on the right side of the vertex (x > h) and slope m = - a on the left side of the vertex (x < h). The graph of f (x) = - a| x - h| + k is an upside-down "V" with vertex (h, k), slope m = - a for x > h and slope m = a for x < h.

Suppose a > 0, then the lowest y-value for y = a| x - h| + k is y = k. If a < 0, then the greatest y-value for y = a| x - h| + k is y = k.

Solve An Absolute Value Equation

If you want to solve an absolute value equation, you have to isolate the absolute value on one side of the equation. Then the next task would be its contents, which have to be equal to both the positive and negative value of the number on the other side of the equation and solve both equations.

Example 1

Solve | x | + 2 = 5.

Isolate the absolute value.

Set the contents of the absolute value portion equal to +3 and –3.

Answer: 3, –3

Example 2

Solve 3| x – 1| – 1 = 11.

Isolate the absolute value.

Set the contents of the absolute value portion equal to +4 and –4.

Solving for x,

Answer: 5, –3

Using absolute values to determine resistance

The error is the deviation of the measured value from the actual value.

Error (E) = Am – At

Am = Measured value

At = True Value

You have to understand the value of Electrical parts to determine the resistance, such as resistors and capacitors come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same.

The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often

±1%,

5%,

10%

Suppose we have a resistor rated at 680 ohms,

Use the absolute value function to express the range of possible actual resistance values.

We can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so with the resistance

in ohms,

−

680

≤

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Examples of how to find the inverse of absolute value functions

If you don't care if the inverse is a function, then f−1(x)=2±x, with the restriction x≥0

If you want the inverse to be a cell function, you must consider a restricted interval initially.

Explanation:

If f is a function that maps x to y, then f−1 is a function that maps y to x. However, if f(x) maps more than one x value to a single y, then there is no way to know which x f−1 should map y to.

You may don't care about the inverse. That includes a function; for that specific reason, we can just consider all the cases, similar to how we invert y=x2 by swapping y and x and then solving for y: x=y2⇒y=±√x. Doing so for f(x)=|2−x|, we get

x=|2−y|

⇒x=±(2−y)

⇒±x=2−y

⇒y=2±x

So, the inverse is y=2±x. Note that as f(x)≥0 in the initial function, we must have x≥0 for the inverse. This gives us the graph, which is a reflection of y=|2−x| across the line y=x, As you would expect

If we want to have the inverse as a function, we must consider a restriction of |x−2| to a domain where every x value is mapped to a distinct y value (a function with this property is called one-to-one).

Again, this is analogous to finding the inverse of g(x)=x2. Suppose we add the initial restriction x≤0, we get x=−√x2⇒g−1(x)=−√x. If we add the initial restriction x≥0, we get x=√x2⇒g−1(x)=√x. Notice that we get a different inverse function depending on how we restrict the domain of g(x).

Because f(x) is defined as

f(x)={−(x−2) ifx−2≤0x−2ifx−2≥0

⇒f(x)= {2−xifx≤2x−2ifx≥2

There are natural restrictions that would be considered the values such as f(x)=|2−x|, x≤2 or f(x)=|2−x|, x≥2. Then we are just finding the inverse of 2−x or x−2 on the restricted interval.

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Frequently Asked Questions By Students

Q1. How do you find the absolute value of a function?

Ans: The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign.

If you take an example, the absolute value of 5 is 5, and the total value of −5 is also 5. So, you can consider the absolute value of a number as its distance from zero along a real number line. Furthermore, the absolute value mainly differentiates the two real numbers, which is the distance between them.

Q2. What does the absolute value function do?

Ans: Taking the absolute value of a negative number actually makes it positive. For this specific reason, graphs of absolute value functions tend not to look quite like the graphs of linear functions that you have already studied.

Q3. What's an absolute value equation?

Ans: The absolute value equations are definite equations that involve expressions with the absolute value functions. This wiki intends to demonstrate and discuss the problem-solving technique that lets us solve some specific equations.