whats the range?

Question

f(x)=|x-2|-7

Answer 1

Hello Olivia!

I'll try and help you out with this problem!

We want to analyze the range of the function below ("range" is basically what "y-values" the function ever reach for all possible input x-values)...

f(x)= |x - 2| - 7

Let's start out by looking at the "base" version of this function and seeing how the applied transformations influence its the range...

"Base" Function:

f(x)= |x|

If we plotted this function, it looks like a perfect "V" shape that has its point exactly on the origin (0,0). Based on this, we can see that the range of f(x)= |x| can be expressed as [ 0 , ∞ ), so f(x) can "reach" y-values as low as 0, and up to +∞.

Note: The '[' symbol means that the exact value of '0' is within the range, whereas a symbol of '(' would mean that the exact value of '0' is not contained within the range.

Now, what transformations have been applied to this "Base" function?

1st Transformation Applied:

f(x)= |x - 2|

This is a horizontal translation of the base function -- moving the base function +2 to the right. Since the "range" is a description of the y-values that a function "reaches", this transformation does not change the range at all from the "base" function.

Now let's apply the 2nd transformation...

2nd Transformation Applied:

f(x)= |x - 2| - 7

This -7 shifts the function downward in the -y direction by 7. This does impact the range, since now the minima (the "point" of the "V" shape) now reaches lower than before -- down to -7.

Now the range of this final function can be expressed as [ -7 , ∞)

Hope that helps!

--Zach

Answer 2

Try thinking about the equation as it would be graphed with linear coordinates. The slope-intercept form of the equation, y=mb+c, can help you visualize the range of this function. Also, splitting the absolute value part into two equalities will help you find the minimum or maximum y-value:

1. y=(x-2)-7
2. y=x-9
3. y=-(x-2)-7
4. simplified, y=-x+2-7
5. y=-x-5
6. setting the two equalities equal to each other gives: x-9=-x-5
7. simplified, 2x=4
8. x=+2

To visualize analytically rather than graphically, substitute x=2 for the original absolute value equation:

1. f(2)=|2-2|-7
2. f(2)=-7
3. y=-7

Since 2 is the x-value with an inflection point, we can tell that the y-value at that point must be a minimum because f(x)=|x-2| does not allow any negative solutions for y. When x=2, y=-7 so:

1. the range for this absolute value equation must be [-7, ∞).

Try graphing this equation on a calculator (Ex. Desmos online graphing calculator) to verify this solution.

Answer 3

l x-2 l ≥ 0

So, l x - 2 l - 7 ≥ -7

Range: set of all real numbers ≥ -7 = [ -7, ∞)