Help with inverse functions...

[traderjoe9]

Suppose g(x) = 8x^9 + 7x^3. Evaluate 8(g^-1(5))^9 + 7(g^-1(5))^3 - 3.

Where it says "g^-1," it's referring to the inverse of function g. I'm ok with other inverse function problems but I can't solve this one...can someone show me step by step how to solve this?

Thanks

 

[mmm4444bot]

This problem looks like an exercise in the meaning of inverse functions.

Start with understanding g[sup:195pa8ye]-1[/sup:195pa8ye](5)

The inverse-function of g is the function that turns g's output back into its originating input.

X ? g(x) ? 5

5 ? g[sup:195pa8ye]-1[/sup:195pa8ye](x) ? X

In other words, X is the value of g[sup:195pa8ye]-1[/sup:195pa8ye](5)

They asked you to evaluate the expression 8(X)^9 - 7(X)^3

Do you see ? That expression is exactly g(X), which is exactly the same as g[g[sup:195pa8ye]-1[/sup:195pa8ye](x)]

So, we simply input X into g(x), and we already know that the output is 5.

You could confirm this, if you're allowed to use technology.

X is about 0.81289 (I graphed and rounded)

In other words:

0.81289 ? g(x) ? g(0.81289) = 5

5 ? g[sup:195pa8ye]-1[/sup:195pa8ye](x) ? g[sup:195pa8ye]-1[/sup:195pa8ye](5) = 0.81289

Therefore, we have:

8(g[sup:195pa8ye]-1[/sup:195pa8ye][5])^9 + 7(g[sup:195pa8ye]-1[/sup:195pa8ye]5])^3 =

8(0.81289)^9 + 7(0.81289)^3