Example Of Inverse Function With Solution

Understanding the example of inverse function is a fundamental skill in algebra that bridges the gap between a function and its reverse mapping. Day to day, this concept not only strengthens algebraic reasoning but also appears frequently in calculus, physics, and real-world problem-solving. When a function takes an input and produces an output, its inverse undoes that process, returning the original input. In this article, we will walk through clear, step-by-step examples with complete solutions, ensuring you can confidently find and verify inverse functions on your own Surprisingly effective..

What Is an Inverse Function?

A function ( f ) has an inverse, denoted as ( f^{-1} ), if and only if it is one-to-one, meaning each output corresponds to exactly one input. The inverse function essentially reflects the original function across the line ( y = x ). If ( (a, b) ) lies on the graph of ( f ), then ( (b, a) ) lies on the graph of ( f^{-1} ). This symmetry is not just visual; it is algebraic. To find an inverse, we typically swap ( x ) and ( y ) in

the equation ( y = f(x) ) and then solve for ( y ). Think about it: the resulting expression represents ( f^{-1}(x) ). On the flip side, this algebraic manipulation is only valid if the original function passes the Horizontal Line Test—no horizontal line intersects the graph more than once—guaranteeing the inverse is also a function.

Step-by-Step Procedure for Finding an Inverse

1. Verify the function is one-to-one. If it is not, restrict the domain so that it becomes one-to-one.
2. Replace ( f(x) ) with ( y ). This makes the equation easier to manipulate.
3. Swap ( x ) and ( y ). This step reflects the input-output relationship.
4. Solve for ( y ). Isolate ( y ) using algebraic operations.
5. Replace ( y ) with ( f^{-1}(x) ). This denotes the inverse function.
6. Verify. Check that ( f(f^{-1}(x)) = x ) and ( f^{-1}(f(x)) = x ) for all ( x ) in the appropriate domains.

Example 1: Linear Function

Find the inverse of ( f(x) = 3x - 5 ).

Solution:

1. This is a linear function with a non-zero slope, so it is one-to-one.
2. Write ( y = 3x - 5 ).
3. Swap variables: ( x = 3y - 5 ).
4. Solve for ( y ): [ x + 5 = 3y \implies y = \frac{x + 5}{3} ]
5. The inverse is ( f^{-1}(x) = \frac{x + 5}{3} ).
6. Verification: [ f(f^{-1}(x)) = 3\left(\frac{x + 5}{3}\right) - 5 = x + 5 - 5 = x ] [ f^{-1}(f(x)) = \frac{(3x - 5) + 5}{3} = \frac{3x}{3} = x ] Both compositions return ( x ), confirming the result.

Example 2: Rational Function

Find the inverse of ( f(x) = \frac{2x + 1}{x - 3} ).

Solution:

1. The function is a rational function where the degree of the numerator equals the degree of the denominator. It is one-to-one on its domain (( x \neq 3 )).
2. Write ( y = \frac{2x + 1}{x - 3} ).
3. Swap variables: ( x = \frac{2y + 1}{y - 3} ).
4. Solve for ( y ): [ x(y - 3) = 2y + 1 \implies xy - 3x = 2y + 1 ] Collect ( y ) terms: [ xy - 2y = 3x + 1 \implies y(x - 2) = 3x + 1 ] [ y = \frac{3x + 1}{x - 2} ]
5. The inverse is ( f^{-1}(x) = \frac{3x + 1}{x - 2} ). Note the domain restriction: ( x \neq 2 ) for the inverse, which corresponds to the horizontal asymptote ( y = 2 ) of the original function.
6. Verification (Composition): [ f(f^{-1}(x)) = \frac{2\left(\frac{3x+1}{x-2}\right) + 1}{\left(\frac{3x+1}{x-2}\right) - 3} = \frac{\frac{6x+2 + x - 2}{x-2}}{\frac{3x+1 - 3x + 6}{x-2}} = \frac{7x}{7} = x ]

Example 3: Radical Function with Restricted Domain

Find the inverse of ( f(x) = \sqrt{x - 4} + 2 ).

Solution:

1. The square root function is inherently one-to-one on its domain. Domain of ( f ): ( x \geq 4 ). Range of ( f ): ( y \geq 2 ).
2. Write ( y = \sqrt{x - 4} + 2 ).
3. Swap variables: ( x = \sqrt{y - 4} + 2 ).
4. Solve for ( y ): [ x - 2 = \sqrt{y - 4} ] Since the principal square root is non-negative, we require ( x - 2 \geq 0 ), or ( x \geq 2 ) (matching the range of ( f )). Square both sides: [ (x - 2)^2 = y - 4 \