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Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.
f(x) = x^3 + 4 and g(x) = cube root of (x4)
 one year ago
 one year ago
Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = x^3 + 4 and g(x) = cube root of (x4)
 one year ago
 one year ago

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ZeHanzBest ResponseYou've already chosen the best response.0
So, first try f(g(x)), this means put the result of function g in function f:\[f(g(x))=\left( \sqrt[3]{x4} \right)^3+4=...\]Can you see what the next step is?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
What is the definition of the 3rd root of a number?
 one year ago

mitchelsewbaranBest ResponseYou've already chosen the best response.0
The number whose cube is equal to a given number
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
OK, but not only a given number, but the number from which you wanted to know the 3rd root: e.g: the 3rd root of 8 is 2, because: 2^3=8. So: the 3rd root of x4 is p, then p³=x4. So...\[(\sqrt[3]{x4})^3=x4\]by definition.
 one year ago

mitchelsewbaranBest ResponseYou've already chosen the best response.0
so that will get the expression out from under the radical
 one year ago

mitchelsewbaranBest ResponseYou've already chosen the best response.0
so i'll get x4+4
 one year ago

mitchelsewbaranBest ResponseYou've already chosen the best response.0
@ZeHanz am i correct?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
You get x4+4=x in my book...
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
So we've done the first half of the proof. Now the second: put the result of f in g:\[g(f(x))=\sqrt[3]{x^3+44}\]
 one year ago

mitchelsewbaranBest ResponseYou've already chosen the best response.0
ok. thnx. but where did u get p from?
 one year ago

ZeHanzBest ResponseYou've already chosen the best response.0
p is not important. You can name it anyway you want. Just look at it this way: say you've got a terribly complex expression, such as:\[3x^{23}5x^{16}+\frac{ 1 }{ 2x }\] First, they want you to calculate the 3rd root of it. I really don't know what it would be, so let's call it p for now:\[p=\sqrt[3]{3x^{23}5x^{16}+\frac{ 1 }{ 2x }}\]Looks awful, not? Now you have to calculate .... p³. Well, you don't have to think long about it, youve got your original complex expression back! The reason for this, that the 3rd root and the 3rd power are each other's inverse. This means: they undo each other's effect on numbers you put in!
 one year ago
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