Problem 61 - Algebra A Combined Function

Question

Find the inverse of each one-to-one function. Then graph the function and its inverse in a square window. $$ f(x)=\sqrt[3]{x+3} $$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = x^3 - 3 \).

Step 1: Set Function as Equation

To find the inverse of the function, start by writing it as an equation with "y" in place of the function, i.e., \( y = \sqrt[3]{x + 3} \).

Step 2: Swap Variables

As the definition of inverse function involves switching the roles of x and y, swap the x and y in the equation to get \( x = \sqrt[3]{y + 3} \).

Step 3: Solve for New Function

Solve for \( y \) in terms of \( x \) from \( x = \sqrt[3]{y + 3} \). Start by cubing both sides to eliminate the cube root, which results in \( x^3 = y + 3 \).

Step 4: Isolate y

Subtract 3 from both sides to isolate \( y \), yielding \( y = x^3 - 3 \).

Step 5: Write Inverse Function

Replace \( y \) with \( f^{-1}(x) \) to express the inverse function as \( f^{-1}(x) = x^3 - 3 \).

Step 6: Graph the Function and its Inverse

To graph \( f(x) = \sqrt[3]{x+3} \) and \( f^{-1}(x) = x^3 - 3 \), note that both graphs should reflect about the line \( y = x \). Use a graphing tool or plot by calculating points. The original function will curve up and the inverse will curve down.

Key Concepts

One-to-One Functions

A one-to-one function is a special type of function where each output value corresponds to exactly one input value. This unique characteristic makes it possible for a function to have an inverse.
When learning about inverse functions, identifying whether a function is one-to-one is a crucial first step.

• For a function to be one-to-one, each horizontal line should intersect the graph of the function at most once. This is known as the Horizontal Line Test.
• If a function passes this test, it means that no two input values produce the same output, and the function has an inverse.
• The inverse function essentially "reverses" the operations of the original function, swapping roles of inputs and outputs.

So, by establishing that a function is one-to-one, we open up the potential to find and meaningfully interpret its inverse. In our original problem, the function \( f(x) = \sqrt[3]{x + 3} \) is one-to-one over its domain, which is all real numbers. Thus, it passes the Horizontal Line Test, making it possible to confidently find its inverse.

Graphing Functions

Graphing functions provides a powerful visual representation of mathematical relationships. In this context, understanding the graphs of both a function and its inverse helps reinforce their connection.

• The graph of the original function \( f(x) = \sqrt[3]{x + 3} \) depicts a standard cubic root curve, shifted horizontally by 3 units due to the \( x+3 \) inside the cube root.
• The graph of the inverse \( f^{-1}(x) = x^3 - 3 \) is a cubic function shifted down by 3 units.
• One vital property of inverse functions is that their graphs are reflections of each other across the line \( y = x \). This means each point \((a, b)\) on the original function corresponds to a point \((b, a)\) on its inverse.

By plotting both graphs, students can more intuitively grasp how functions and their inverses behave relative to each other. This skill is significant not only for visualizing mathematics but also for confirming the correctness of algebraic work on finding inverses.

Cubic Functions

Cubic functions are polynomial functions of degree three, typically expressed in the form \( f(x) = ax^3 + bx^2 + cx + d \). They are fundamental in mathematics due to their distinct shape and properties.

• These functions have a characteristic "S" or "N" shaped curve. Depending on the coefficient \(a\), the curve either rises or falls steeply, before leveling out, and then rising or falling steeply again.
• Cubic functions can have one real root or three real roots, and their graphs can feature inflection points where the curve changes concavity.
• In our exercise, the function \( f^{-1}(x) = x^3 - 3 \) is such a cubic function. It presents a single inflection point, reflects symmetry typical of cubic graphs, and displays internal axes shifts due to the \(-3\).

Understanding cubic functions is crucial because they often surface in real-world phenomena, modeling situations involving volume, growth, and more. When we come across cubic functions in algebra, geometry, or calculus, recognizing their properties helps in analyzing and sketching their graphs accurately.