Natural Log Functions

[oliviafaithp]

So this is the problem

Given the function f defined by f(x)= ln(x[sup:3gl2cffs][2][/sup:3gl2cffs] -9)
a) Determine the symmetry of the graph of f. Justify
I found that there will be symmetry over the y-axis. How should I justify?
b) Find the domain of f.
I know that x cannot equal -3, +3, ,-2,+2, -1,+1, or zero, but how do I write this?c) Find all values of x such that f(x)=0
?
d) Write a formula for the inverse function of f, for x>3.
I know to switch the x and y and solve for y, but how do you cancel out a natural log?

 

[mmm4444bot]

Hi Olivia:

For (a), we show that the output does not change if we change the sign on the input (i.e., the value of y is the same regardless on which side of the Origin x lies).

In other words, we show that f(x) = f(-x).

For (b), you excluded seven Real numbers from the domain, but, actually, the number of values that must be excluded is infinite!

If you're trying to say that x cannot be any number from -3 through +3, then you're correct.

There are different ways to report this. Here's set-builder notation.

Domain of f : {x is Real | x < -3 or x > 3}

We could also write the union of these two sets using interval notation.

Domain of f : (-?, -3) U (3, ?)

For (c), we need to solve the following equation, which we get after substituting the number zero for the variable f(x).

ln(x^2 - 9) = 0

Next, we use the logarithm definition to switch to exponential form.

e^0 = x^2 - 9

Now solve in the usual way.

For (d), we also first switch to exponential form, the same way.

e^x = y^2 - 9

After you solve for y, don't forget to consider the restricted domain on f. In other words, one of the expressions for y must be excluded.

If I wrote anything that you don't understand, then please reply with specific questions.

Otherwise, show whatever work you can, if you would like more help with any of this, so that we can see where to continue helping you.

Cheers,

~ Mark