How do you graph the function #f(x)=x^3-2# and its inverse?

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Answer 1 · 2017-07-04T01:57:51

See explanation

To graph #f(x)=x^3-2# it is helpful to remember what the graph of #f(x)=x^3# looks like (See below):

#f(x)=x^3#

graph{x^3 [-10, 10, -5, 5]}

To graph #f(x)=x^3-2#, all we really doing is shifting the #x#-values down #2# units.

#f(x)=x^3-2#

graph{x^3-2 [-10, 10, -5, 5]}

As for graphing we first must find the inverse, #f^(-1)(x)#. We do this by switching the roles of #x# and #y# and solving for #y.

We can rewrite #f(x)=x^3-2# as #y=x^3-2# which then becomes #x=y^3-2# when we switch the roles of #x# and #y#

Now, we solve for #y#

#x=y^3-2#

#x+2=y^3+cancel(-2+2#

#root(3)(x+2)=root(3)(y^3)#

#root(3)(x+2)=y#

Our inverse function can then be rewritten as:

#f^(-1)(x)=root(3)(x+2)#

To graph #f^(-1)(x)=root(3)(x+2)#, take the function of #f(x)=root(3)x# and shift the #x#-values #2# units to the left:

#f(x)=root(3)(x+2)#

graph{root(3)(x+2) [-10, 10, -5, 5]}

The graph of the function is below along with the original function:

graph{(y-x^3+2)(y-root(3)(x+2))=0 [-10, 10, -5, 5]}