Finding Inverse Exponential Functions: A Method for Solving Without Answers

In summary, the conversation discusses finding the values of a and b for the exponential function f(x) = ab^x, given f(2) = 2 and f(4) = 18. The solution involves solving for a and b using the given equations, and then finding the inverse function by plugging in the known values and using logarithms.
  • #1
omg precal
33
0
First of all, sorry for asking so many questions. I do not want answers, just a method of solving them.

Homework Statement



For the exponential function f(x) = ab^x, suppose f(2) = 2 and f(4) = 18.

a. Find a and b.
b. Find f^-1(54), the inverse function.

Homework Equations



None, really...

The Attempt at a Solution



a. f(2) = 2, meaning ab^2 = 2. f(4) = 18, meaning ab^4 = 18.

(ab^2 = 2) * 9 -> 9ab^2 = 18
ab^4 = 18

9ab^2 = ab^4

divide both sides by ab^2

9 = b^2
b = 3

a(3)^2 = 2
9a = 2
a = 2/9

b. The inverse of ab^54...

Inverse of: (2/9)(3)^54

And from here, I'm lost. Do you fifty-fourth root everything because it is the inverse?
 
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  • #2
inverse

omg precal said:
b. Find f^-1(54), the inverse function.
This means: find the value of x such that ab^x = 54.
 
  • #3
that's what the inverse does to the function?
 
  • #4
that gives me 3^x = 243. using logarithms...

x log3 = log 243

x = log 243/ log 3

x = 5

am i correct?
 
  • #5
Plug your answer into the original equation and see for yourself! :smile:
 
  • #6
3^5 is indeed 243.

Thanks, Doc Al.
 
  • #7
One thing I remember about the inverse function is that it's the function that for a value of y, would return the x-value that the original function used.

f(x)
x | y
-----
1 | 3
2 | 12
3 | 27
4 | 48
5 | 75

So for
f^-1(x)
x | y
-----
3 | 1
12 | 2
27 | 3
48 | 4
75 | 5

So:

y = ab^x
Solve for x to get the inverse function:

x = lnb / (lny - lna)

I think. I'm not sure at all.
 

1. What is an inverse exponential function?

An inverse exponential function is a mathematical relationship between two variables, where one variable is exponentially related to the other. It is the reverse of an exponential function, where the independent variable is the exponent.

2. How do you find the inverse of an exponential function?

To find the inverse of an exponential function, you need to switch the positions of the dependent and independent variables and solve for the new dependent variable. This can be done by taking the logarithm of both sides of the original function.

3. What is the domain and range of an inverse exponential function?

The domain of an inverse exponential function is the range of the original exponential function, and the range of an inverse exponential function is the domain of the original exponential function.

4. How do you graph an inverse exponential function?

To graph an inverse exponential function, you can use the same techniques as graphing any other function. Start by creating a table of values, then plot the points on a graph and connect them with a smooth curve. Remember to label the axes and include any necessary transformations.

5. What are some real-life applications of inverse exponential functions?

Inverse exponential functions can be used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. They are also commonly used in fields such as economics, physics, and biology to analyze and predict exponential relationships between variables.

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