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Ewan M.

Calculus 3

1 month, 2 weeks ago

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For each function $f(x)$ given, prove (using a composition) that $g(x)=f^{-1}(x)$ $$f(x)=x^{2}+8 ; x \geq 0, g(x)=\sqrt{x-8}$$

Chapter 4

Exponential and Logarithmic Functions

Section 1

One-to-One and Inverse Functions

so in this problem were given to functions. F of X equals X squared minus eight and g of X equals X minus the square root of X minus. Say we need to show that G of X is the inverse of f of axe When we do that, using the definition of an inverse which says that the inverse of the normal function of acts equals acts. Then we plug in f of X into G of X. We get the square root, we have a domain Restriction X is greater than or equal to zero, which will be important later. So we plug in f of x and G of X. We get squared of f of X minus eight on equals X And then we plug in what we have here for f of X so squared of X squared minus Oh, and that's actually a plus eight here. So x squared plus eight minus a minus eight equals X. You could see plus and minus eight cancel and we get square of cancel and we get square root of X squared equals acts and we can see this wouldn't work if we plug in negative seven Then we would get negative seven equals for the square root of 49 which is taken to be seven. So negative seven is not equal to seven. But since we can't have a negative X since X is greater than or equal to zero, then this is true. We can do this. We can simplify the square root out so g of X is the inverse of f of X.

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