what is inverse of this funtion
f(x)=(1 / 3)^x?
I think the answer is
f-1(x)= log base 1/3 of x, but i know im wrong... please help!!! THANK YOU
Three steps to finding the inverse:
1. interchange x and y.
2. solve for y in terms of x.
3. evaluate f-1(f(x)) to verify that it gives x as the result.
0. y=(1/3)^x
1. x=(1/3)^y
2. log(x)=ylog(1/3)
=> y=log(x)/log(1/3)=-log(x)/log(3)
or
f-1(x)=-log(x)/log(3)
3.
y(f(x))
=y((1/3)^x)
=-log((1/3)^x)/(log(3)
=-xlog(1/3)/log(3)
=x OK
THANKS ALOT... I GET IT NOW!!!!!!
You're welcome!
To find the inverse of the given function f(x) = (1/3)^x, you are on the right track. Let's go over the steps to find the correct answer.
Step 1: Start with the given function: f(x) = (1/3)^x.
Step 2: Replace f(x) with y: y = (1/3)^x.
Step 3: Swap x and y: x = (1/3)^y.
Step 4: Solve for y. To isolate y, you need to take the logarithm of both sides of the equation. However, be careful with the base of the logarithm you choose. The choice of base will affect the form of the inverse function. In this case, we will use the natural logarithm with base e (ln):
ln(x) = ln((1/3)^y).
Step 5: Apply the logarithmic properties. Using the property that ln(a^b) = b * ln(a), we can rewrite the equation as:
ln(x) = y * ln(1/3).
Step 6: Solve for y. Divide both sides of the equation by ln(1/3):
y = ln(x) / ln(1/3).
So, the correct inverse function is:
f^(-1)(x) = ln(x) / ln(1/3).
Note that this is the natural logarithm of x over the natural logarithm of (1/3).