Exponential Growth Functions

  • #1
139
0

Homework Statement



Consider the exponential function f(x)= ab^(x)

a) Show that (f(x+1))/f(x) = b

b)Use the results from part (a) to explain why there is no exponential function of the form f(x)= ab^(x) whose graph passes through the points (0,4) (1,4) (2,8) (3,24) and (4,72)

Homework Equations



-None-

The Attempt at a Solution



There is no attempt at part (a) becuase I couldn't find a way to start it, but for part (b) I am thinking it can't be a exponential function becuase the y-values for two of the x-values are the same. That's what I think, except I can't put it togethor the way part (b) wants me to.

Please Help.
 

Answers and Replies

  • #2
eumyang
Homework Helper
1,347
10
Consider the exponential function f(x)= ab^(x)

a) Show that (f(x+1))/f(x) = b
You are given
[itex]f(x) = ab^x[/itex]
Are you familiar with function notation? What goes inside the parentheses next to f can indicate what you're plugging in. For example,
[itex]f(5) = ab^5[/itex]
[itex]f(\text{Mickey Mouse}) = ab^\text{Mickey Mouse}[/itex]
Given that, what is
[itex]f(x+1) ?[/itex]
Once you find that, then substitute into
[tex]\frac{f(x+1)}{f(x)}[/tex]
b)Use the results from part (a) to explain why there is no exponential function of the form f(x)= ab^(x) whose graph passes through the points (0,4) (1,4) (2,8) (3,24) and (4,72)
Suppose (0, 4) is on the graph of f(x). This is saying that
[itex]f(0) = 4[/itex]
Suppose the other points are on the graph of f(x) as well. What is
[itex]f(1) ?[/itex]
[itex]f(2) ?[/itex]
...
Can you see how this relates to the expression
[tex]\frac{f(x+1)}{f(x)} ?[/tex]
 
  • #3
139
0
Thanks, I know get part (a), I was misunderstanding the problem before. It has to be like (ab^(x+1))/(ab^(x)) and then you show simplification. :) :)

Sorry about that whole thing, I understand it now, but how do I take this and put it with part (b)?
 
Last edited:
  • #4
eumyang
Homework Helper
1,347
10
I got this: ((ab^(x))+1)/(ab^(x))

what do i do from there, if thats correct?

I don't really get what your saying but I do know what functions are and that whatever value you have for f(x) you plug into all x variables.
If you mean
[tex]\frac{ab^x + 1}{ab^x}[/tex]
... then sorry, that's not right. Is the "+1" really supposed to be separate from abx?
 
  • #5
eumyang
Homework Helper
1,347
10
Thanks, I know get part (a), I was misunderstanding the problem before. It has to be like (ab^(x+1))/(ab^(x)) and then you show simplification. :) :)

Sorry about that whole thing, I understand it now, but how do I take this and put it with part (b)?
In (b), in order for the points to lie on the graph of f(x) = abx, then for any x,
[itex]\frac{f(x+1)}{f(x)}[/itex]
has to equal the same base b. So what is
[itex]\frac{f(1)}{f(0)} ?[/itex]
[itex]\frac{f(2)}{f(1)} ?[/itex]
And so on.
 
  • #6
139
0
I don't understand what you are saying...

The question:

b)Use the results from part (a) to explain why there is no exponential function of the form f(x)= ab^(x) whose graph passes through the points (0,4) (1,4) (2,8) (3,24) and (4,72)

From the points, which is the base? I understand you take the points and plug it into the equation, but I don't get anything further then that. For example, for the point (0,4). I would do this:

y=b^x
4=b^0
4≠1

Is that right to show that it is not a function becuase the values are unequal.

^^Thats probably wrong but what do I plug into the (f(x+1))/(f(x)) =b ????? <--- Thats where I am getting confused
 
Last edited:
  • #7
139
0
bumping this becuase its the last problem and I am getting irritated :(
 
  • #8
eumyang
Homework Helper
1,347
10
From the points, which is the base? I understand you take the points and plug it into the equation, but I don't get anything further then that. For example, for the point (0,4). I would do this:

y=b^x
4=b^0
4≠1
Why did you plug into y = bx? The function is in the form of y = abx. Anyway, that's not what I would do.

Let's look at another exponential function: y = 5*2x. Some of the points are these:
(0, 5), (1, 10), (2, 20), (3, 40), (4, 80)...
For an exponential function f, and for any x,
[itex]\frac{f(x+1)}{f(x)} = b[/itex].
Take the ratio of successive y-values, like thus:
[itex]\frac{f(1)}{f(0)} = \frac{10}{5} = 2[/itex]
[itex]\frac{f(2)}{f(1)} = \frac{20}{10} = 2[/itex]
[itex]\frac{f(3)}{f(2)} = \frac{40}{20} = 2[/itex]
[itex]\frac{f(4)}{f(3)} = \frac{80}{40} = 2[/itex]
...
Note that they all equal 2, and that is the base of the exponential function y = 5*2x.

Now look at the points you were given:
(0,4) (1,4) (2,8) (3,24) and (4,72)
and find
[itex]\frac{f(1)}{f(0)} = ?[/itex]
[itex]\frac{f(2)}{f(1)} = ?[/itex]
[itex]\frac{f(3)}{f(2)} = ?[/itex]
[itex]\frac{f(4)}{f(3)} = ?[/itex]
Do these simplify to the same number?
 

Related Threads on Exponential Growth Functions

Replies
1
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
2
Views
6K
  • Last Post
Replies
9
Views
982
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
4
Views
1K
Replies
4
Views
784
Top