How do you find the inverse of functions? for example how would you find the inverse of y= log8 x
Let's try this for f(x)= y = e^(6x)
Step 1:
interchange x and y to get
x=e^(6y)
Step 2:
solve for y in terms of x:
ln(x) = ln(e^(6y)) = 6y
y = ln(x)/6
So
f-1(x) = ln(x)/6
Step 3:
Verify that f(f-1(x))=x (if the inverse was correct).
f(f-1(x))
=f(ln(x)/6)
=e^(6*ln(x)/6)
=e^(ln(x))
=x
So the inverse is correct.
Ok that's slightly confusing.
I understand the flipping of x and y but then what do you do?
so x=log8 y
?
Yes, then you solve for y in terms of x.
use the law of exponents:
elog(x)=x
or
8log8 y=y, etc.
(assuming log8 y is log(y) to the base 8)
raise both sides to power of 8 to get
8x = 8log8(y)<?sup>
Simplify and solve y in terms of x.
8x = 8log8(y) = y
Simplify and solve y in terms of x.
MMMMmmmmmmk that helps, thanks!
You're welcome!
To find the inverse of a function, you can follow these steps:
1. Start with the given function, in this case, y = log8(x).
2. Replace the given function with y, so the equation becomes x = log8(y).
3. Swap the positions of x and y, making the equation y = log8(x).
4. Solve the equation for y. In this case, let's isolate y by getting rid of the logarithm. Rewrite the equation in exponential form: 8^y = x.
5. Take the logarithm base 8 of both sides: log8(8^y) = log8(x).
6. The logarithm base 8 of 8 to the power of y simplifies to just y: y = log8(x).
7. The resulting equation, y = log8(x), is the inverse of the given function, y = log8(x).
So, the inverse of y = log8(x) is y = log8(x).
Remember that the inverse of a function undoes the original function. In this case, applying the inverse function of logarithm with base 8 to the output of the original function will return the original input.