Find the inverse of each function f(x) given, then prove (by composition) your inverse function is correct. Note the domain of f is all real numbers.

Question

Find the inverse of each function f(x) given, then prove (by composition) your inverse function is correct. Note the domain of f is all real numbers.

Answer 1

To find the inverse, solve for the other variable. In this case, x=? To solve this, f(x) or y needs to become the independent variable and X must become the dependent one.

First let's get rid of the root by cubing both sides. y^3=3x-2.

Get rid of the constant by adding to both sides. y^3+2=3x

Then solve for x by dividing both sides by the coefficient 3. x=(y^3+2)/3.

Now we put what we've gotten for x in where x is in the original equation. (3((y^3+2)/3)-2)^1/3=y. Because the math cancels out and leads to a true statement, the one is an inverse of the other.

Answer 2

Hello Xavier!

I like to solve these problems by changing f(x) for y and the putting x as a function of y.

f(x) = cube root (3x-2)

y = cube root (3x-2)

y³ = 3x -2

3x = y³ + 2

x = (y³ + 2)/3

Let's call the inverse function of f(x) as g(x). By the previous calculation:

g(x) = (x³ + 2)/3

To prove by composition, if we change x in g(x) for f(x) we must get just the variable x.

g(f(x)) = (3x - 2 + 2)/3 = x

And also, if we change x in g(x) for g(x) we must get just the variable x.

f(g(x)) = cube root (x³ + 2 - 2) = x

So we prove that g(x) and g(x) are one the inverse of the other.

Answer 3

Step 1: Replace f(x) with y

y = cuberoot(3x - 2)

Step 2: Switch x with y and y with x

x = cuberoot(3y - 2)

Step 3: Solve for "y"

x = cuberoot(3y - 2) [cube both sides of the equation to get:

x³ = 3y - 2 [ Add to to both sides to get:

x³ + 2 = 3y [Divide both sides by 3 to get:

(x³ + 2)/3 = y

Step 4: Replace "y" with f^-1(x):

f^-1(x) = (x³ + 2)/3

Proof of the fact that this is the inverse function by composition:

Show f(f^-1(x)) = x and f^-1(f(x)) = x

f(f^-1(x)) = cuberoot{3[(x³ + 2)/3] - 2}

= cuberoot(x³ + 2 - 2)

= cuberoot(x³)

= x

f^-1(f(x)) = {[cuberoot(3x - 2)]³ + 2)/3

= (3x - 2 + 2)/3

= 3x/3

= x