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Explain why the inverse of f(x)=x^2 is not a function, but the inverse of g(x)=x^3 is a function.

1 Answer

2 votes
Answer:

Because its graph will not pass the vertical line test.

Explanation:

We must first find the inverse of the given functions.

Let


y=f(x).

Then for the first function, we have
y=x^2

We interchange x and y to get,


x=y^2

We make y the subject to get,


\pm √(x)=y

This is not a function because one x-value is mapping onto two y-values.

Hence its graph will not pass the vertical line test.
See red graph.

For the second function, we again let


y=g(x)

Then,


y=x^3

We interchange x and y to get,


x=y^3

We make y the subject to get,


x^{(1)/(3)}=y

This is a function because, one x-value maps on to one and only one y-value. This tells us that the graph of this function will past the vertical line test.

see blue graph.
Explain why the inverse of f(x)=x^2 is not a function, but the inverse of g(x)=x^3 is-example-1
Explain why the inverse of f(x)=x^2 is not a function, but the inverse of g(x)=x^3 is-example-2
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