How do you find the inverse of #f(x) = 5x^3 - 7#?

1 Answer
May 16, 2018

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Inverse of the function #color(red)(f(x)=5x^3-7# is given by #color(blue)(root(3)((x+7)/5#

Explanation:

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Note:

The Inverse of a function may NOT always be a function.

So, if the inverse of a function is a function by itself, then it is called an Inverse Function.

How do we determine these inverse relationships ?

Method 1

We can find the inverse of the function by simply swapping the ordered pairs.

Method 2

(a) Set the function to #y#

(b) Swap the #x#, #y# variables

(c) Solve for #y#

Method 3

The graph of an inverse function is the reflection of the original graph over the line #color(red)(y=x#, called the Identity Line.

#color(green)("Step 1 : "#

Given the function : #color(red)(y=f(x)=5x^3-7#

Construct a data table for this function and graph it.

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Behavior of the Parent Graph shown

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#color(green)("Step 2 : "#

We use the Method 2 to solve

(a) Set the function to #y#

#y= 5x^3-7#

(b) Swap the #x#, #y# variables

#x=5y^3-7#

(c) Solve for #y#

We have, #x=5y^3-7#

Subtract #5y^3# from both sides.

#rArr x-5y^3=5y^3-7-5y^3#

#rArr x-5y^3=cancel (5y^3)-7-cancel(5y^3)#

#rArr x-5y^3=-7#

Subtract #x# from both sides

#rArr x-5y^3-x=-7-x#

#rArr cancel x-5y^3-cancel x=-7-x#

Multiply both sides by #(-1)# to remove negative signs.

#rArr (-1)(-5y^3)=(-1)(-7-x)#

#rArr 5y^3=7+x#

#rArr 5y^3=x+7#

Divide both sides by #5#

#rArr (5y^3)/5=(x+7)/5#

#rArr (cancel 5y^3)/cancel 5=(x+7)/5#

#y^3=(x+7)/5#

Take Cube Root on both sides

#rArr root(3)(y^3)=root(3)[(x+7)/5]#

Cube and the Cube Root cancel each other.

#rArr y=root(3)[(x+7)/5]#

Hence,

Inverse of the function #color(red)(f(x)=5x^3-7# is given by #color(blue)(root(3)((x+7)/5#

#color(green)("Step 3 : "#

Explore the graph:

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