Determine domain and codomain for f(x) = ln(x^2 - 6x)

The domain:

Interval:

Inequality: x<0 or x>6

Words: x is less than 0 or greater than 6.

The range:

Interval:

Inequality:

Words: All real numbers.

Step-by-step explanation:

For the natural log (Ln) to exist, the inside must be positive.  

So the domain can be found by solving the following:

Let's factor:

is a faced up parabola with x-intercepts x=0 and x=6.  This means it is positive when x<0 or when x>6.  Those are the parts with the curve of the parabola is above the x-axis.

So the domain is x<0 or x>6.

Interval notation if you prefer would be: .

as equivalent exponential form .

So let's look at the parabola one more time....

has it's minimum occur halfway between the x-intercepts we found earlier. Parabolas are symmetric about their axis of symmetry which the vertex lays on. So the halfway point of x=0 and x=6 is x=3.  The vertex occurs at x=3.

To find the corresponding y-coordinate we can replace x with 3:

.

The lowest point is -9 since the parabola is opened up.

So we know that .

We also know every exponential function is greater than 0 so the intersections of what I just mentioned about our and all exponential functions is just >0.

for all

So the range is all real numbers.

As an inequality:

As an interval: .