Question #7d077

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Answer 1 · 2018-01-11T17:27:27

See below.

A function is one to one if:

No two elements in the domain map on to the same element in the range.

Since we have a restricted domain, this criteria is met, so $f(x)$ is one to one

For an unrestricted domain $f(x)=x^2$ is a many to one function.

Inverse of $f(x)$

We need to express $x$ as a function of $y$ :

$y=x^2$

$x=+-sqrt(y)$

Substituting $x=y$

$y=+-sqrt(x)$ $:.$ $f^-1(x)=+-sqrt(x)$

Domain of $f(x)$

For $x<=0$

Domain is ${x in RR: -oo < x<=0}$

Since $x^2>=0$ for all $RR$

Range is:

${y in RR:0 <=y< oo}$

Domain of $f^-1(x)$

Because the domain of $f(x)$ is $x<=0$ , only the inverse $y=-sqrt(x)$ is needed. The domain of this will be the same as the range of $f(x)$ i.e.

${x in RR: 0 <= x< oo}$

The range will be the same as the domain of $f(x)$ i.e.

${y in RR : -oo < y <= 0 }$

For one to one functions the range of $f(x)$ is the domain of $f^-1(x)$ and the domain of $f(x)$ is the range of $f^-1(x)$ .