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9.

 

P(x)   =   55000√[ x - 1945 ]

 

In which year was the population of the city  220,000 ? That is, what is  x  when  P(x) = 220,000 ?

 

220,000   =   55000√[ x - 1945 ]

                                                      Divide both sides of the equation by  55000

4   =   √[ x - 1945 ]

                                 Square both sides of the equation.

16   =   x - 1945

                                 Add  1945  to both sides.

1961   =   x

 

Check:    P(1961)   =   55000√[ 1961 - 1945 ]   =   55000√[ 16 ]   =   55000(4)   =   220,000

10.

 

f(x)  =  \(\sqrt[3]{x-4}\)

                               Instead of  f(x)  write  y .

y   =   \(\sqrt[3]{x-4}\)

                               Raise both sides to the power of  3 .

y³   =   x - 4

                               Add  4  to both sides.

y³ + 4  =   x

                               Now we have solved the original function for  x , so the inverse function is...

f^-1(x)   =   x³ + 4

 

If  f(x)  and  f^-1(x)  are inverses, then on a graph they should be a reflection of each other about the line y = x .  Here's a graph to check that:  https://www.desmos.com/calculator/oeidolt8op