. Show your work.2/3/4. 136.151: Test #1
20 minutes
Name: _____________________ Student Number: _______
1. Find an equation of the line passing through a point (2,3) and parallel to the line . Show your work.
Solution:
The line through (2,3) and with slope m has an equation 
. So, we need to find m. But we know that the required line is parallel to the line ; consequently they have equal slopes. Now note that the later equation is equivalent to 
; so the slope of that line is -1. Thereby the slope m that we need is also -1. So the required equation is 
2. Find the center and the radius of the circle 
.
Solution.
is equivalent to (means the same as ) 
(to check that simply expand the square in the last equation). The last equation is obviously equivalent to 
. We now see that the center of the circle is at (0,-3) and that its radius is 
.
3. Find the domain of the function 
.
Solution.
Since the denominator should not be equal to 0, we must have 
and 
. Since the usual 
is defined only for number that are 
, we must have 
, i.e., 
. We can now summarize: the domain of the function 
consists of all numbers x at most equal to 2 except 
.
4. Show that if the functions and 
are both odd functions then their product 
is an even function.
Solution.
We assume here that both and
are odd. This means that 
and that 
for every x. We want to
show that the function 
is even, i.e.
we want to show that 
. Let's see:
,
where in the second step we have used that both and are odd.