Describe the following function: f(x) = 3 - |2x - 5|

[Math_Junkie]

Describe the following function: f(x) = 3 - |2x - 5|

I'm having problems with absolute value functions, any help would be appreciated.

 

[tkhunny]

One can ALWAYS decode absolute values by eliminating the absolute values. First, you must understand the fundamental principle.

|x| = x if x >= 0 Examples |0| = 0, |3| = 3

Here we are saying that the absolute value does nothing for positive numbers and for zero. If you are sure the argument is positive or zero, simply discard the absolute values and you are done.

|x| - -x if x < 0 Examples |-2| = -(-2) = 2, |-10| = -(-10) = 10

Here we are saying that the function of the absolute value simply changes the sign of negative numbers. If you are sure the argument is negative, simply discard the absolute values and change the sign of the argument.

Having said that, where is 2x-5 positive or zero? 2x-5 >= 0 ==> 2x >= 5 ==> x >= 5/2
Similarly, 2x-5 < 0 for x < 5/2

Now we have two cases.

For x >= 5/2, 3 - |2x-5| = 2 - (2x-5) = 2-2x+5 = 7-2x -- Graph that and erase everything for x < 5/2

For x < 5/2, 3 - |2x-5| = 2 - -[(2x-5)] = 3 + (2x-5) = 2+2x-5 = -3+2x -- Graph that and erase everything for x >= 5/2

That's one way to do it.

 

[soroban]

Hello, Math_Junkie!

Describe the following function: .\(\displaystyle f(x) \:=\:3 - |2x-5|\)

I assume that "describe" means that a graph is sufficient.


\(\displaystyle \text{We're expected to know the graph of: }\:y \:=\:|x|\)

        *       |       *
          *     |     *
            *   |   *
              * | *
      - - - - - o - - - - -
                |

\(\displaystyle \text{The graph of: }\,y \:=\:|2x|\,\text{ is the same, only "steeper".}\)

            *   |   *
             *  |  *
              * | *
               *|*
      - - - - - o - - - - -
                |

\(\displaystyle \text{The graph of: }\,y \:=\:|2x-5|\,\text{ has been moved 5 units to the }right.\)

                |   *       *
                |    *     *
                |     *   *
                |      * *
      - - - - - + - - - o - - -
                |       5

\(\displaystyle \text{The graph of: }\,y \:=\:-|2x-5|\,\text{ is reflected over the x-axis.}\)

                |
                |
      - - - - - + - - - o - - -
                |      * *
                |     *   *
                |    *     *
                |   *       *
                |

\(\displaystyle \text{The graph of }\,y \:=\:-|2x-5| + 3\,\text{ is }raised\,\text{ 3 units.}\)

                |
                |     (5,3)
                |       o
                |      * *
      - - - - - + - - * - * - -
                |    *     *
                |   *       *
                |

\(\displaystyle \text{And }that\text{ is the graph of }\,y \:=\:3 - |2x-5|\)

 

[Math_Junkie]

Thanks so much for the help. Both answers helped immensely.

 

[mmm4444bot]

... I'm having problems with absolute value functions; any help would be appreciated.

Hi Math Junkie:

I would like to add comments to this discussion on how I often think "graphically" (i.e., in terms of a number line) when I think about absolute value, although these comments are not directed specifically toward the exercise posted. (In other words, the following comments are for the future when wondering what absolute value means.) Maybe they will help you some day ...

When we were small, we only knew about zero and numbers bigger than zero.

Then, one day, we discovered THE NUMBER LINE.

We found out that zero is NOT at one end of the number line; it is in the CENTER!

This causes initial confusion when thinking about "big" and "small" numbers.

Which is bigger? -1 or -10,000?

Well, the numerical value of -10,000 is MUCH SMALLER than the numerical value of -1.

But, on the other hand, the distance that you need to move away from zero on the number line in order to get to -10,000 is MUCH BIGGER than the distance that you need to move to reach -1.

In other words, the MAGNITUDE of -10,000 is much larger than the magnitude of -1.

So, with the introduction of negative numbers, and the fact that every real number other than zero has an opposite, we get into situations where we don't need to discuss ON WHICH SIDE of zero a particular number is located, but, rather, HOW FAR AWAY FROM ZERO it is located.

When we do this, we are talking about the number's magnitude, or "absolute" value.

Both of the numbers 4 and -4 are four units away from zero.

Both of the numbers -7/8 and 7/8 are 7/8ths of a unit away from zero.

The distance away from zero (on the number line) is a number's absolute value.

Thinking graphically makes it clear to me why 4 and -4 have the same absolute value, and why -7/8 and 7/8 have the same absolute value.

When you see the mathematical statement y = |x|, it says NOTHING about which side of zero y is located. It only gives you the MAGNITUDE of the quantity y. It tells you that y is x units away from zero. So, we better consider both possibilities if we don't want to miss something!

Consider the following mathematical statement.

|2x - 5| = 3

This tells you that the unknown quantity 2x-5 is located 3 units away from zero on the number line.

If you want to find all of the values of x that make this statement true, then you need to remember to consider BOTH halves of the number line.

In other words, 2x-5 can either be 3 or -3. You need to solve BOTH of the following to discover both solutions.

2x - 5 = 3

2x - 5 = -3

When you see absolute value symbols, try to remember that their presence is there as a statement about magnitude (not numerical value) AND that this magnitude comes from both a positive number and its opposite (unless, of course, this number turns out to be zero).

Cheers,

~ Mark