Problem 45 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(c(x)=4+\sqrt[3]{x}\)

Step-by-Step Solution

Answer

The inverse function is \(c^{-1}(x) = (x - 4)^3\).

Step 1: Set the function equal to y

Start by setting the function notation equal to y. This gives us:\[ y = 4 + \sqrt[3]{x} \]

Step 2: Solve for x.

To find the inverse, we need to solve for \(x\) in terms of \(y\). First, isolate \(\sqrt[3]{x}\) by subtracting 4 from both sides:\[ y - 4 = \sqrt[3]{x} \] Next, cube both sides to eliminate the cube root:\[ (y - 4)^3 = x \]

Step 3: Write the inverse function

Now that \(x\) is expressed in terms of \(y\), rewrite it as an inverse function by swapping \(x\) with \(c^{-1}(x)\):\[ c^{-1}(x) = (x - 4)^3 \]

Key Concepts

Cubic Root

The cubic root is a mathematical operation that identifies a number which, when cubed, results in the original number. In simpler terms, if you want to find the cubic root of a number, you're looking for a value that, when multiplied by itself twice (total three times), equals that number. For example, the cubic root of 27 is 3 because \[ 3 \times 3 \times 3 = 27. \]The notation for cubic root is written as \( \sqrt[3]{x} \). Here, "3" is an indicator of the cube, and \(x\) is the number for which you're finding the cube root.

• The cubic root can be both positive and negative since the cube of a negative number is still negative. For instance, the cubic root of -8 is -2 because \[ (-2) \times (-2) \times (-2) = -8. \]
• Cubic roots are significant because they help in simplifying equations and expressing inverses, especially when dealing with polynomial functions.

Function Notation

Function notation provides a way to denote functions mathematically using symbols and variables to represent a relationship between input and output. It's a compact and efficient method to describe how each input value maps to an output value using a function, like \( f(x) \). In our example, we used \(c(x)\) to signify a specific function.

• The letter "\(c\)" signifies the name given to the function.
• The "\(x\)" within the parentheses is the variable representing the input to the function.

Using function notation helps in easily swapping variables when finding inverses or making calculations. Function notation also enables efficient communication of relationships, making it easier to discover such inverse functions. This is crucial for solving equations and transforming expressions.

Solving Equations

Solving equations involves finding the value of an unknown variable, often represented as \(x\) or \(y\), that makes the equation true. To solve an equation, you need to perform operations that simplify the equation so that the variable has a clear solution.In the context of our problem, the goal was to "flip" the function to express \(x\) in terms of \(y\). Here are some fundamental steps for solving equations:

• Isolating the variable: Start by performing arithmetic operations that isolate the variable on one side of the equation. For our equation \( y = 4 + \sqrt[3]{x} \), this process begins by subtracting 4 from both sides, giving \( y - 4 = \sqrt[3]{x} \).
• Eliminating radicals: To remove the cubic root, cube both sides, which results in \( (y - 4)^3 = x \).
• Finding inverses: Once isolated, express the solved variable in function notation as the inverse function, such as \(c^{-1}(x) = (x - 4)^3\).

Solving equations is a valuable skill in mathematics that enables you to find exact solutions and understand relationships between variable quantities.