Question

range of f(x) =-|x-3|

Answer 1

Short Answer: Range of f(x) = -|x-3|

=> { f(x) | f(x) ≤ 0 }

Generally, if the graph of the function sits on the x-axis, its range is less than or equal to 0.

Let's have f(x) = |x|

Right now, it sits on the origin point as there is nothing stating it shouldn't.

If you change it to f(x) = |x-1|, the x coordinate of the function moves to the right by one point.

On the other hand, if you change it to f(x) = |x+2|, the x coordinate of the function moves to the left by two points.

To change the position of the abs. value function vertically(its y coordinate), you do the same stuff but outside of the absolute value.

I.e.

f(x) = |x| sits on the origin point.

If f(x) = |x| + 1, the vertex of the graphed function moves up by one.

If f(x) = |x| - 3, the vertex of the graphed function moves down by 3.

EXAMPLE USING ALL OF THE ABOVE RULES:

f(x) = |x-2| + 3 would sit the vertex of the function at: (2, 3).

LAST POINT:

Now, you might be wondering: Where should i point the opening of the function graph?

Think of it like the symbols < and > but rotated so that they face up and down.

If f(x) = |x|, the opening faces up.

If f(x) = -|x|, the opening faces down.

LAST EXAMPLE USING ALL OF THE ABOVE RULES:

f(x) = -|x + 1| - 5

The vertex of the abs. value function would lie on (-1, -5) and the opening would point down since |x| is negative.

That's all!

Remember, inside the absolute value lines changes the x-axis position of the vertex,

Outside the abs. value lines will change the y-axis position of the vertex,

And a positive abs. value will point the opening of the graph up while a negative abs. value will point it down.

Cya!