For each of the piecewise-defined functions in determine whether or not the function is one-to-one, and if it is, determine its inverse function. f(x)={x2when x&lt;0xwhen x≥0

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For each of the piecewise-defined functions in determine whether or not the function is one-to-one, and if it is, determine its inverse function. f(x)={x2when x&amp;lt;0xwhen x≥0

yunitsiL · Accepted Answer

Step 1 Let x&amp;lt;0 Then we get, f(x)=x2 Let us consider two numbers x&amp;lt;0andy&amp;lt;0. Also let us assume, x2=y2 x2−y2=0 (x + y)(x - y) = 0 (x + y) = 0, (x - y) = 0 x = -y, x = y Therefore for two elements x and -x we have f(x)=x2, as well as f(−x)=(−x)2=x2=f(x). Hence f is not one-to-one for x&amp;lt;0. Step 2 Let x≥0 Then we get, f(x) = x Let us assume for two elements x and y both greater than or equal to zero, their images are same. So we get, f(x) = f(y) x = y So for x≥0,y≥0, f(x) = f(y) implies x = y. Hence f is one-to-one function.

For each of the piecewise-defined functions in determine whether or not the function is one-to-one, and if it is, determine its inverse function.f(x)={(x^2, when x < 0),(x, when x>=0):}

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