1. ## Graph/Domain/Range?

I'm totally fried right now guys, I feel like I haven't seen equations like these in years!

The domain of the function f(x)= 4-x is?

The range of the function f(x)= 2- x^2 is?

And finally...
The graph of the function f(x)= 1/ (x^2-9):
has two vertical asymptotes?
is symmetric about the x axis but not about the y axis?
has two horizontal asymptotes?
is symmetric about the line x=3?

2. Originally Posted by lemon301
I'm totally fried right now guys, I feel like I haven't seen equations like these in years!

The domain of the function f(x)= 4-x is?

The range of the function f(x)= 2- x^2 is?

And finally...
The graph of the function f(x)= 1/ (x^2-9):
has two vertical asymptotes?
is symmetric about the x axis but not about the y axis?
has two horizontal asymptotes?
is symmetric about the line x=3?
Some definitions:

The domain of a function is the set of possible values for the independent variable(s). In this case we are looking for the set of possible values for x. Generally it is easier to find what values of x are not possible, then give the answer as all real numbers not including the impossible values. So are there any values of x such that f(x) = 4 - x does not exist?

The range of a function is a little more difficult. The range of a function is the set of values that the function returns. (ie. what values can f(x) take?) My best recommendation is to graph the function (which is a downward opening parabola) and see what values of the function are allowed. So is there a lower limit for what f(x) = 2 - x^2 can be? Is there an upper limit?

For vertical asymptotes we are looking for values of x such that the denominator is 0.

For horizontal asymptotes we are looking for what the behavior of the function is for x tending toward positive and negative infinity. If the function approximates a constant for these limits, then it has a horizontal asymptote.

For symmetry about the x axis we are looking to see if for every point (x, f(x)) on the graph we also have the point (x, -f(x)) on the graph.

For symmetry about the line x = 3, the simplest thing to do is first consider y = f(x - 3). This means translate the function three units to the left. Then the question becomes is f(x - 3) symmetric about the line x = 0, aka the y axis? If a function g(x) is symmetric about the y axis then g(-x) = g(x). So the question, in its final form, becomes: Is f(-(x - 3)) = f(x - 3)?

Try your question with these and post what you come up with if you are still having problems.

-Dan

3. Originally Posted by topsquark
Some definitions:

The domain of a function is the set of possible values for the independent variable(s). In this case we are looking for the set of possible values for x. Generally it is easier to find what values of x are not possible, then give the answer as all real numbers not including the impossible values. So are there any values of x such that f(x) = 4 - x does not exist?

The range of a function is a little more difficult. The range of a function is the set of values that the function returns. (ie. what values can f(x) take?) My best recommendation is to graph the function (which is a downward opening parabola) and see what values of the function are allowed. So is there a lower limit for what f(x) = 2 - x^2 can be? Is there an upper limit?

For vertical asymptotes we are looking for values of x such that the denominator is 0.

For horizontal asymptotes we are looking for what the behavior of the function is for x tending toward positive and negative infinity. If the function approximates a constant for these limits, then it has a horizontal asymptote.

For symmetry about the x axis we are looking to see if for every point (x, f(x)) on the graph we also have the point (x, -f(x)) on the graph.

For symmetry about the line x = 3, the simplest thing to do is first consider y = f(x - 3). This means translate the function three units to the left. Then the question becomes is f(x - 3) symmetric about the line x = 0, aka the y axis? If a function g(x) is symmetric about the y axis then g(-x) = g(x). So the question, in its final form, becomes: Is f(-(x - 3)) = f(x - 3)?

Try your question with these and post what you come up with if you are still having problems.

-Dan

Thanks so much for the help! Only thing I'm having trouble with is domain. Right now I'm looking at an equation f(x)= [square root] 16 - x^2.
I'm trying but I keep thinking different answers. I know it has to be [0,4] [-2,2] [0,16] or [-4,4]. I have the feeling if I keep trying I'm just going to come up with different answers lol.

4. Originally Posted by lemon301
Thanks so much for the help! Only thing I'm having trouble with is domain. Right now I'm looking at an equation f(x)= [square root] 16 - x^2.
I'm trying but I keep thinking different answers. I know it has to be [0,4] [-2,2] [0,16] or [-4,4]. I have the feeling if I keep trying I'm just going to come up with different answers lol.
It's domain is from -4 to 4. The radical cannot be negative, so solve for such an inequality...

$16 - x^2 \ge 0$

$x^2 \le 16$

$-4 \le x \le 4$