Find the function inverse ,f-1(x) , of y = 2x + 1.
The Correct Answer and Explanation is:
To find the inverse function ( f^{-1}(x) ) of the function ( y = f(x) = 2x + 1 ), follow these steps:
Step 1: Replace ( f(x) ) with ( y )
Start by rewriting the function in terms of ( y ):
[
y = 2x + 1
]
Step 2: Solve for ( x )
To find the inverse, we need to express ( x ) in terms of ( y ). Start by isolating ( x ):
- Subtract 1 from both sides:
[
y – 1 = 2x
] - Divide both sides by 2:
[
x = \frac{y – 1}{2}
]
Step 3: Switch ( x ) and ( y )
Since we are looking for the inverse function, we replace ( y ) with ( x ):
[
f^{-1}(x) = \frac{x – 1}{2}
]
Conclusion
Thus, the inverse function is:
[
f^{-1}(x) = \frac{x – 1}{2}
]
Explanation of the Process
Finding the inverse of a function essentially involves reversing the roles of the input and output. The original function ( y = 2x + 1 ) transforms ( x ) values into ( y ) values through a linear relationship. To derive the inverse, we start by expressing the function as an equation in ( y ) and then isolate ( x ) to express it in terms of ( y ).
By rearranging the equation, we find ( x ) as a function of ( y ). In this case, we subtracted 1 from both sides to eliminate the constant term on the right side and then divided by 2 to isolate ( x ). The crucial step is to remember to switch ( x ) and ( y ) once we have isolated ( x ).
The resulting inverse function ( f^{-1}(x) = \frac{x – 1}{2} ) indicates that for any ( x ) value, we can find the original input of the function ( f(x) ) by first subtracting 1 and then dividing the result by 2. This method can be applied to any one-to-one function, ensuring that each output corresponds to exactly one input, allowing for the proper definition of an inverse function.