Problem 19 - College Algebra Essentials

Question

The functions in Exercises \(11-28\) 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 \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\) $$f(x)=(x+2)^{3}$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{x} - 2\), and both \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\) hold true.

Step 1: Finding the inverse of the function

Firstly, replace \(f(x)\) with \(y\), the equation becomes \(y = (x+2)^{3}\). To find the inverse, exchange \(x\) and \(y\), leading to \(x = (y+2)^{3}\). Now, solve for \(y\) to get the inverse function. The cube root of \(x\) is equal to \(y+2\). Substract \(2\) from both sides to isolate \(y\), which results \(y = \sqrt[3]{x} - 2\). So, \(f^{-1}(x) = \sqrt[3]{x} - 2\).

Step 2: Verifying the inverse function Part I

To verify the found inverse function, substitute \(f^{-1}(x)\) into \(f(x)\) such that \(f(f^{-1}(x))\). Substituting \(f^{-1}(x) = \sqrt[3]{x} - 2\) into \(f(x) = (x+2)^{3}\) yields \(f(f^{-1}(x)) = ((\sqrt[3]{x} - 2) + 2)^{3}\). After simplification, the equation becomes \(f(f^{-1}(x)) = (\sqrt[3]{x})^{3}\) which equals to \(x\).

Step 3: Verifying the inverse function Part II

Now, substitute \(f(x)\) into \(f^{-1}(x)\) such that \(f^{-1}(f(x))\). Substituting \(f(x) = (x + 2)^{3}\) into \(f^{-1}(x) = \sqrt[3]{x} - 2\) yields \(f^{-1}(f(x)) = \sqrt[3]{(x+2)^{3}} - 2\). Simplifying it results in \(f^{-1}(f(x)) = x + 2 - 2 = x\). Hence, the equation \(f^{-1}(f(x)) = x\) has been satisfied.

Key Concepts

One-to-One Functions

A one-to-one function is a specific type of function where each input is paired with a unique output. This means no two different inputs map to the same output.
This unique pairing is key for ensuring that an inverse function can exist. In other words, a function is one-to-one if different inputs produce different outputs.

• For every value of \(x_1\) and \(x_2\), if \(x_1 eq x_2\), then \(f(x_1) eq f(x_2)\).
• It passes the Horizontal Line Test: A horizontal line drawn through the graph of the function will intersect the graph at most once.

When a function is one-to-one, it means that its inverse function will also be a valid function. Understanding one-to-one functions is crucial when dealing with inverse functions because if a function isn't one-to-one, its inverse won't meet the definition of a function.

Verification of Inverse Functions

To verify that one function is indeed the inverse of another, you need to perform specific operations to ensure their compositions yield expected results.
Specifically, the following two conditions must be satisfied:

• For the function \(f\) and its inverse \(f^{-1}\), \(f(f^{-1}(x)) = x\) should hold true for every \(x\) in the domain of \(f^{-1}\).
• Also, \(f^{-1}(f(x)) = x\) should be true for every \(x\) in the domain of \(f\).

In simpler terms, applying the function after its inverse (or vice versa) should just give you the input you started with. This cyclic property helps in checking that the inverse and the original function are indeed correct. As illustrated in the provided exercise, we checked these conditions by simplification, confirming that both conditions hold, thereby verifying the accuracy of the inverse function.

Cubic Functions

Cubic functions are polynomial functions of degree three. They take the form \(f(x) = ax^3 + bx^2 + cx + d\). Each cubic function graph has a distinct characteristic look, often possessing a sort of 'curved' shape.
They can have up to three real roots and create an "S" shape in the Cartesian plane, depending on coefficients.

• Cubic functions can have one, two, or three real roots.
• Their graphs are continuous and smooth, without breaks or sharp angles.
• They can be strictly increasing, decreasing, or have inflection points (points where the direction of curvature changes).

For cubic functions, especially one-to-one cubic functions like \(f(x) = (x+2)^3\), inverses are attainable. Such functions pass the Horizontal Line Test, making them suitable for inverse operations. Finding an inverse involves solving the equation in terms of \(x\), ensuring the steps are reversible and accurate. Cubics serve as excellent introductions to higher-level polynomial functions, with inverse relationships demonstrating elegant algebraic concepts.