# 1.6 Absolute value functions

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In this section you will:
• Graph an absolute value function.
• Solve an absolute value equation.
• Solve an absolute value inequality.

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

## Understanding absolute value

Recall that in its basic form $\text{\hspace{0.17em}}f\left(x\right)=|x|,\text{\hspace{0.17em}}$ the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.

## Absolute value function

The absolute value function can be defined as a piecewise function

$\text{\hspace{0.17em}}f\left(x\right)=|x|=\left\{\begin{array}{ccc}x& \text{if}& x\ge 0\\ -x& \text{if}& x<0\end{array}\text{\hspace{0.17em}}$

## Determine a number within a prescribed distance

Describe all values $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ within or including a distance of 4 from the number 5.

We want the distance between $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and 5 to be less than or equal to 4. We can draw a number line, such as the one in [link] , to represent the condition to be satisfied.

The distance from $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to 5 can be represented using the absolute value as $\text{\hspace{0.17em}}|x-5|.\text{\hspace{0.17em}}$ We want the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that satisfy the condition $|x-5|\le 4.$

Describe all values $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ within a distance of 3 from the number 2.

$|x-2|\le 3$

## Resistance of a resistor

Electrical parts, 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 $\text{\hspace{0.17em}}\text{±1%,}\text{\hspace{0.17em}}±\text{5%,}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}±\text{10%}\text{.}$

Suppose we have a resistor rated at 680 ohms, $\text{\hspace{0.17em}}±5%.\text{\hspace{0.17em}}$ Use the absolute value function to express the range of possible values of the actual resistance.

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 $\text{\hspace{0.17em}}R\text{\hspace{0.17em}}$ in ohms,

$|R-680|\le 34$

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

using the variable $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ for passing, $\text{\hspace{0.17em}}|p-80|\le 20$

## Graphing an absolute value function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in [link] .

[link] shows the graph of $\text{\hspace{0.17em}}y=2|x–3|+4.\text{\hspace{0.17em}}$ The graph of $\text{\hspace{0.17em}}y=|x|\text{\hspace{0.17em}}$ has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at $\text{\hspace{0.17em}}\left(3,4\right)\text{\hspace{0.17em}}$ for this transformed function.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this