Given f(x)=8x^5, find f^-1(x)
then state whether f^1(x) is a function.

f^-1(x) = (x/8)^(1/5) is a function

Step-by-step explanation:

The inverse of a function is the function reflected across the line y=x. This results in the coordinate points (x,y) of the function becoming (y,x) for the inverse function. Algebraically to find the inverse, switch the x and y locations and solve for y.

y = 8x^5

x = 8y^5

x/8 = y^5

(5)√(x/8) = y

This is a fifth root of (x/7) also written in exponents as y = (x/8)^(1/5).

This is a function. While it appears not to be a function because the middle portion over the origin appears vertical, it is a function because the middle portion over the origin is changing and graphing software shows it has no input with more than one output. Without graphing software you would rule it is not a function.

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: