Find the inverse the function and the domain and the range of f and f−1, if the function is f(x)=2xx+5 and it's one-to-one.

Question

Find the inverse the function and the domain and the range of f and f−1, if the function is f(x)=2xx+5 and it&#039;s one-to-one.

Sally Cresswell · Accepted Answer

To find the inverse function, substitute x=f−1andy=x and calculate:  y=2xx+5  xy+5y=2x  x(y−2)=−5y  x=−5y2−y  x=5y2−y  f−1(x)=5x2−x  If f(f−1(x))=f−1(f(x)),  then f−1(x) is an inverse function of f(x)  =f(f−1(x))  =f(5x2−y)  =2×5x2−y5x2−x+5  =10x2−x5x+10−5x2−x  =x  The domain of f(x) is range for f−1(x)  And, the range of f(x) is domain for f−1(x)  f(x)=2xx+5  x+5=0  x=-5  Domain =(−∞,−5)∪(−5,∞)  f−1(x)=5x2−x  2-x=0  x=2  Range =(−∞,2)∪(2,∞)

Find the inverse the function and the domain and the range of f and f−1,...

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