4 Major topics that i have understood:
Logs: I understand how to solve logarithms. I learned that when given a problem, (for instance log base 2 of 32), i am actually looking for the power. In this example, the answer would be 5, since 2^5 = 32. I also learned how to solve more complex logarithms, such as log base 6 of 1/36^1/5. In this case, the answer would be -2/5, considering 6^-2/5 is 1/36^1/5.
Inverses: I understand the term one-to-one. In order for a function to be one-to-one, it must pass the horizontal line test. This means that the graph of the line must not intersect a horizontal line at more than one point. This test is indeed similar to the vertical line test done to check if a graph is a function. However, the horizontal line test deals with the use of a horizontal line, of course.
Inverses: I understand how to find the inverse of a function. To find the inverse graphically, i learned that the graph of f(x) is simply reflected across the line y=x. The line that results is the graph of f^-1(x). In this line, the original values of x and y are simply switched to form the graph of the inverse of f(x). To find the inverse of a function algebraically, i learned that you take the following steps:
- Replace the x with a y.
- Isolate the y variable.
- Divide by the number in front of the y variable (if necessary) in order to get 1y (as opposed to 2y, 5y, etc).
- Change y=something into f^-1(x)= something. This form is the right form for your answer and is much more "telling" as Ms. Hwang says.
Example: find the inverse of f(x) = 2x +3.
f(x) = 2x +3
x = 2y +3
-3 -3
x-3 = 2y
x-3/2 = 2y/2
x-3 = y
f^-1(x) = x-3
Inverses: I understand that if f^-1(x) is really the inverse of f(x), then f(f^-1(x)) =x and f^-1(f(x))=x. This is very useful when using algebra to find the inverse of a function. After solving for the inverse, i learned to check my answers by solving f(f^-1(x)) and f^-1(f(x)). If i get x as an answer both times, then i know i found the correct inverse.
What i did not understand completely:
I did not understand how to graph logarithm functions. For example, i didn't know how to do numbers 39-42 of homework C2. I know you can use a graphing calculator to graph these, but i want to learn how to graph these equations myself.
I also did not know how to solve problems 35 & 36 of homework C2.
I did not understand how f(g(x))=x but yet, g(x) is not the inverse of x. (# 53 of HW C3)
I also did not understand how to solve # 60 from HW C3.
If anyone knows how to solve these, please help me:] I would very much appreciate your effort!
Rocio and Jesus helped me with #35, which is similar to 36 as well.
ReplyDeleteDo the reciprocal of e ^ -x to the original equation given to you.
Then, multiply both sides by e^x in order to get 1/e^x to become "1". Now you should have e^2x+1 = 3e^x.
Next you subtract 3e^x to both sides and get e^2x-3e^x+1= 0.
Now its in the format of a quadratic equation.
You then have to complete the square
Treat e^x as just x
After completing the square you get
3 +/- (square root) 5 / 2.
"After that you take the natural log of everything to cancel out the e and you have your answer"
To graph logs you need to change them to exponential form which is something like 2^x=y.Once you have it in that form you can graph them no problem.
ReplyDeletei made a post about graphing logarithms on my blog if you want to see how.
ReplyDelete#53...eh, i don't have a definite answer, but i'll share what i've come up with. we are asked whether g(x) is the inverse of f(x) given that f(g(x))=x. at first, i myself said yes and was shocked to discover the answer is false. after some thinking, i came up with this: in the given information, it is assumed that the inverse, g(x), is a function. but based soley from the given information, we can't conclude the inverse is a function as well (in short, there is no way to conclude f(x) is one-to-one). they use f(x)=x^2 and g(x)=root of x as an example because it fits into f(g(x))=x, but g(x) can't be written in that form because it isn't a function perhaps it's some form of a proof by contradiction?...). based on that, if g(x) isn't a function, f(g(x))=x can't be true because the statement assumes both are functions. again, not completely sure if this is true, so i'd ask Ms. Hwang just in case.
for #60, also not 100% sure, but i would assume that if a function is one-to-one, its reciprocle would have the same properties?...
i did #60 but to tellyou the truth im not sure if i did it right or if i just proved it half way but i can probably help you see it a diffferent way:
ReplyDeleteso
g(x)= 1/f(x) move f(x) to other side
g(x)f(x)=1
so now you could say:
f(x)=1/g(x)
It doesnt mean that g(x)= f(x)
But you can figure that g(x) has to equal to 1 and f(x) has to equal to 1 so that when multiply them they give you 1
Not only does multiplying 1 and 1 give you 1 but multiplying the x (being any number) and 1/x (the reciprical of the number) give you 1
I doubt i answered the question but i pointed something out right??