Finding an Inverse Function : 1. Use the Horizontal Line Test to decide whether f has an inverse function. 2. In the equation for f(x), replace f(x) by y. 3. Interchange the roles of x and y and solve for y. 4. Replace y by f - 1(x) in the new equation. 5. Verify that f and f - 1 are inverse functions of each other by showing that the domain of f is equal to the range of f - 1, the range of f is equal to the domain of f - 1, and f ( f - 1(x) ) = x and f - 1( f(x) ) = x.
\The function is f(x) = - x2 + 9.
\(1). Applying the Horizontal Line Test :
\The graph of the function given by f(x) = - x2 + 10 is shown in the above Figure. Because it is possible to find a horizontal line that intersects the graph of f at more than one point, you can conclude that f is not a one-to-one function and does not have an inverse function.
\Finding Inverse Functions Informally :
\Find the inverse function of f(x) = - x2 + 10. Then verify that both f ( f - 1(x) ) and f - 1( f(x) ) are equal to the identity function.
\The original function f(x) = - x2 + 10.
\Replace f(x) by y.
\y = - x2 + 10
\Interchange x and y.
\x = - y2 + 10
\Solve for y.
\x - 10 = - y2
\- x + 10 = y2
\y = ± √(- x - 9).
\Replace y by f - 1(x).
\f - 1(x) = ± √(- x - 9).
\The ± indicates that to a given value of x there correspond two values of y. So, y is not a function of x.
\The function f(x) = - x2 + 10.
\The above function is quadratic function, so the domain of the function is all real numbers.
\Domain : x R
\Case 1 : If the domain is x 0 then f - 1(x) = + √(- x - 9)
\The domain of the function is all real numbers
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