Problem 28 - College Algebra

Question

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=\frac{2 x-3}{x+1}$$

Step-by-Step Solution

Answer

\(f^{-1}(x)=\frac{-x-3}{x-2}\)

Step 1: Invert the Function

To find the inverse, replace \(f(x)\) with \(y\), then swap \(x\) and \(y\) and solve for \(y\):Given \(f(x)=\frac{2 x-3}{x+1}\). Let's replace \(f(x)\) with \(y\); now we have \(y=\frac{2 x-3}{x+1}\). Swap \(x\) and \(y\), so our new equation becomes \(x=\frac{2 y-3}{y+1}\). Solve for \(y\), which will be the inverse function, \(f^{-1}(x)\).

Step 2: Solve for the inverse function

Multiply both sides by \((y+1)\) to eliminate the denominator:\(x(y+1)= 2y - 3\).Then distribute the x:\(xy + x = 2y - 3\).Now, put all terms with \(y\) on one side and the constants on the other:\(xy - 2y = -x - 3\).Factor out \(y\) from the left side:\(y(x - 2) = -x - 3\).Finally, divide by \((x - 2)\) to solve for \(y\):\(y= \frac{-x-3}{x-2}\).Therefore, \(f^{-1}(x)=\frac{-x-3}{x-2}\).

Step 3: Verify the inverse function

To verify, check that 1) \(f\left(f^{-1}(x)\right)=x\) and 2) \(f^{-1}(f(x))=x\).1) Start with \(f(f^{-1}(x))\): substituted \(f^{-1}(x)\) into the equation for \(f(x)\) \[f\left(f^{-1}(x)\right) = \frac{2\left(\frac{-x-3}{x-2}\right)-3}{\frac{-x-3}{x-2}+1}\]Simplify the equation, you should get \(x\).2) To show \(f^{-1}(f(x))=x\), substitute \(f(x)\) into the equation for \(f^{-1}(x)\). \[f^{-1}\left(f(x)\right) =\frac{-\left(\frac{2x-3}{x+1}\right)-3}{\left(\frac{2x-3}{x+1}\right)-2}\]Simplify the equation, you should get \(x\). This confirms that we correctly identified the inverse function.

Key Concepts

Understanding One-to-One Functions

A one-to-one function is a function where each output value corresponds to exactly one input value. These functions are essential because only they have unique inverses. If a function is not one-to-one, it might map different inputs to the same output, making it impossible to reverse the process with certainty.
To check if a function is one-to-one, you can use the Horizontal Line Test:

• If any horizontal line crosses the graph of the function more than once, the function is not one-to-one.

A one-to-one function has an inverse function, which "undoes" the original function. Thus, understanding and identifying one-to-one functions is crucial when dealing with inverses.

Exploring Composition of Functions

Composing functions involves the process of combining two functions where the output of one function becomes the input of another. In mathematical terms, the composition of functions \(f\) and \(g\) is denoted by \((f \circ g)(x)\), and means \(f(g(x))\).
When it comes to inverse functions, composition is a useful tool for verification. By composing a function with its inverse, you should end up with the original input value. This property is crucial because it demonstrates that the functions are truly inverse to one another.

• For any function \(f(x)\) and its inverse \(f^{-1}(x)\), the composition \(f(f^{-1}(x))\) should return \(x\).
• Similarly, the composition \(f^{-1}(f(x))\) should also return \(x\).

These equations are key indicators that you're working with a correctly identified pair of inverse functions.

Verifying Inverse Functions

Verifying inverse functions is an essential step to ensure accuracy in calculations. The process involves demonstrating that applying the inverse function to the original function successfully returns the original input. This includes calculating both \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) and checking their equivalences with \(x\).
The procedure:

• Substitute the expression for \(f^{-1}(x)\) into \(f(x)\).
• Simplify the resulting expression. You should simplify it to \(x\) if the functions are truly inverses.

Next, repeat this process the other way:

• Substitute the expression for \(f(x)\) into \(f^{-1}(x)\).
• Again, simplify; if correct, you'll get \(x\).

This verification is a foolproof method to check that the inverse was found and applied correctly, although simplifying these expressions may require careful algebraic manipulation.