There are two WebWork exercises that go beyond what we’ve covered in class this semester (although you should’ve seen this material in MAT1275 or an equivalent algebra course). Hence I will count these two exercises as extra credit, but thought I’d also provide some hints to help you solve them:
Functions – Translations: Problem 4: In this exercises, you are given at equation of the form
$latex x^2 + y^2 = r^2$
The graph of this equation is a circle centered at the origin (0,0) with radius r.
In general, the equation of a circle centered at a point (h,k) with radius r is (x-h)^2 + (y-k)^2 = r^2 .
Use that to figure out the solution to the WebWork exercise, given that you are asked to shift the circle, and hence shift the center of the graph from (0,0) to another point (h,k).
Functions – Inverse Functions: Problem 10: In this exercise you are given a function f(x) involving e^(2x) terms, and you are asked to find the inverse function f^{-1}(x).
In general, as we outlined in class and as you should’ve done on the previous exercises in this HW set, in order to solve for the inverse function, set up the equation y = f(x), and then solve for x in terms of y.
In order to do that in this exercise, where the x terms are in the exponent, you need to use the natural log function.
Here’s a simpler example:
Say f(x) = 7e^{2x}. Then in order to find f^{-1}(x):
y = 7e^{2x}
y/7 = e^{2x}
and now take the natural log of both sides:
ln (y/7) = ln (e^{2x})
On the LHS, ln (e^{2x}) = 2x (this is by the definition of ln, which is the inverse function of e^x!)
Hence,
ln (y/7) = 2x
x = ln (y/7)/2
and so in this example,
f^{-1}(x) = ln (x/7)/2
