Problem 26 - Intermediate Algebra

Question

Find the inverse of the given function by using the "undoing process," and then verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). (Objective 4) $$f(x)=-\frac{2}{3} x$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = -\frac{3}{2}x\), and both compositions result in \(x\).

Step 1: Write the Function as an Equation

Start with the function given, which is \(f(x) = -\frac{2}{3}x\). To find the inverse, we first rewrite this as \(y = -\frac{2}{3}x\).

Step 2: Swap Variables

Interchange \(x\) and \(y\) to help find the inverse. So, the equation becomes \(x = -\frac{2}{3}y\).

Step 3: Solve for y

Solve the equation \(x = -\frac{2}{3}y\) for \(y\). Multiply both sides by \(-\frac{3}{2}\) to isolate \(y\):\[ y = -\frac{3}{2}x \]

Step 4: Write the Inverse Function

Now, with \(y\) expressed as \(y = -\frac{3}{2}x\), the inverse function can be written as \(f^{-1}(x) = -\frac{3}{2}x\).

Step 5: Verify the Composition (f ∘ f inverse)

To verify \((f \circ f^{-1})(x) = x\), substitute \(f^{-1}(x)\) into \(f(x)\):\(f(f^{-1}(x)) = f\left(-\frac{3}{2}x\right) = -\frac{2}{3}\left(-\frac{3}{2}x\right) = x\).

Step 6: Verify the Composition (f inverse ∘ f)

To verify \((f^{-1} \circ f)(x) = x\), substitute \(f(x)\) into \(f^{-1}(x)\):\(f^{-1}(f(x)) = f^{-1}\left(-\frac{2}{3}x\right) = -\frac{3}{2}\left(-\frac{2}{3}x\right) = x\).

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics that involves combining two functions, where the output of one function becomes the input of another. This is often written as \(f \circ g\left(x\right)\). In simple terms, you can think of it as plugging one function into another function. For example, if you have functions \(f(x)\) and \(g(x)\), the function composition \(f(g(x))\) means you first apply \(g\), then apply \(f\) to the result.

• Start with an inner function.
• Find the output from that function.
• Use this output as an input for another function.

In the context of finding an inverse, verifying that \(f \circ f^{-1}=x\) or \(f^{-1} \circ f=x\) shows the undoing effect of these functions. This means that when you apply one function and then its inverse, you end up with the original value (x). It demonstrates that these functions systematically reverse each other, maintaining equilibrium.

Undoing Process

The undoing process is a technique used to find the inverse of a function. The principle is to reverse each operation performed in the original function to retrieve the input. This reversal requires algebraic manipulation.

• First, express the function as an equation: \(y = -\frac{2}{3}x\).
• Switch the roles of \(x\) and \(y\) to find potential inverses: \(x = -\frac{2}{3}y\).
• Solve for \(y\) as it's now the subject: multiply by the reciprocal to isolate \(y\).
• Result in an expression like \(y = -\frac{3}{2}x\).

This approach is akin to rewinding a tape to its beginning, undoing every action step by step, similar to clicking "undo" on actions performed. It confirms that every output has a corresponding input, proving the concept of inverse functions.

Solving Equations

Solving equations is key to finding inverse functions, especially during the undoing process. When finding an inverse, solving for the dependent variable yields the key relationship between variables. It involves breaking down operations layer by layer until the desired variable stands alone on one side of the equation.

• Rearrange the equation to put the \(x\) term isolated.
• Use inverse operations: if the original operation was multiplication, use division.
• Apply the correct arithmetic to clear fractions or coefficients.

In this exercise, we solved \(x = -\frac{2}{3}y\) to find \(y\) by multiplying both sides by the reciprocal, \(-\frac{3}{2}\), to obtain \(y = -\frac{3}{2}x\). Once you derive the inverse relation, you can check it by plugging back through composition, ensuring correctness. This systematic methodology is much like solving a puzzle, where every piece must fit perfectly to validate the solution.