`f(x) = |x + 2|, [-2, oo)` Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval.

Friday, November 5, 2010

$\displaystyle{f{{\left({x}\right)}}}={\left|{x}+{2}\right|},{\left[-{2},\infty\right)}$ Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval.

We are asked to show that $\displaystyle{f{{\left({x}\right)}}}={\left|{x}+{2}\right|},{\left[-{2},\infty\right)}$ has an inverse by showing that the function is monotonic on the interval using the derivative:

By definition, $\displaystyle{f{{\left({x}\right)}}}={\left|{x}+{2}\right|}={\left\lbrace\matrix{{x}+{2}&{x}+{2}\ge{0}\\-{x}-{2}&{x}+{2}<{0}}\right\rbrace}$

x+2>0 ==> x>-2 which is the interval we wish so on the interval f(x)=x+2.

f'(x)=1 which is positive for all x in the interval so the function is monotonic (strictly increasing) on the interval. Thus the function has an inverse.

The graph:

Ninenotes

Friday, November 5, 2010

$\displaystyle{f{{\left({x}\right)}}}={\left|{x}+{2}\right|},{\left[-{2},\infty\right)}$ Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval.

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