# Thread: Finding the inverse of a function

1. ## Finding the inverse of a function

Find the inverse of the function:

f(x) = (3/4) x^5 + 5

read "three-fourths x to the fifth plus 5"

Graphing my progress on my calculator as I go, the last point where I get an inverse is at

5^[sqrt] ((4/3)x - (20/3))
^(root with an index of 5)

If I am thinking correctly, you cannot have a root in the denominator, so I've been multiplying the whole thing above by

5^[sqrt](3^4)
^again, root with an index of 5

to get the fifth root out of the denominator, but this messes the whole equation up and it no longer mirrors the original function when I plug it into my graphing calculator. Is there some rule with roots about square roots that I am unaware of?

Help is much appreciated.

2. Originally Posted by happydino1
Find the inverse of the function:

f(x) = (3/4) x^5 + 5

read "three-fourths x to the fifth plus 5"

Graphing my progress on my calculator as I go, the last point where I get an inverse is at

5^[sqrt] ((4/3)x - (20/3))
^(root with an index of 5)

If I am thinking correctly, you cannot have a root in the denominator, so I've been multiplying the whole thing above by

5^[sqrt](3^4)
^again, root with an index of 5

to get the fifth root out of the denominator, but this messes the whole equation up and it no longer mirrors the original function when I plug it into my graphing calculator. Is there some rule with roots about square roots that I am unaware of?

Help is much appreciated.
Analytically to find the inverse of a function $\displaystyle y = f(x)$ we switch the roles of x and y to get $\displaystyle x = f(y)$, then solve for y: $\displaystyle y = g(x)$. g(x) is our inverse function.

So
$\displaystyle y = f(x) = \frac{3}{4}x^5 + 5$

Goes to
$\displaystyle x = \frac{3}{4}y^5 + 5$

$\displaystyle x - 5 = \frac{3}{4}y^5$

$\displaystyle y^5 = \frac{4}{3}(x - 5)$

$\displaystyle y = \sqrt[5]{\frac{4}{3}(x - 5)}$

-Dan