Problem 40 - A Graphical Approach to College Algebra

Question

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=x^{3}-7, \quad g(x)=\sqrt[3]{x+7}$$

Step-by-Step Solution

Answer

\(f(x)\) and \(g(x)\) are inverses because both \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Definition of Inverse Functions

Two functions \(f\) and \(g\) are inverses if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domains of \(f\) and \(g\). We will verify both compositions.

Step 2: Compute \(f(g(x))\)

Start by substituting \(g(x)\) into \(f(x)\): \(f(g(x)) = f(\sqrt[3]{x+7}) = (\sqrt[3]{x+7})^3 - 7\). Simplifying this, we get \(x + 7 - 7 = x\). Hence, \(f(g(x)) = x\).

Step 3: Compute \(g(f(x))\)

Now substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g(x^3 - 7) = \sqrt[3]{(x^3 - 7) + 7}\). Simplifying, we get \(\sqrt[3]{x^3} = x\). Thus, \(g(f(x)) = x\).

Step 4: Satisfy Both Conditions

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\) hold true, the functions \(f\) and \(g\) are indeed inverses according to the definition of inverse functions.

Key Concepts

Composition of Functions

When discussing functions in mathematics, it's important to understand how functions can be combined. The combination process, known as "composition of functions," involves applying one function to the result of another function. If you have two functions, say \( f(x) \) and \( g(x) \), the composition \( f(g(x)) \) means you first apply \( g \) to \( x \), and then apply \( f \) to the result of \( g(x) \). This is a crucial concept because the composition of two functions may yield interesting properties, especially when dealing with inverse functions.

• For two functions \( f \) and \( g \), \( f \circ g \) denotes the composition, which is read as "\( f \) of \( g \)."
• To verify if two functions are inverses, you need to check if both compositions \( f(g(x)) \) and \( g(f(x)) \) return \( x \).
• This characteristic must hold for all \( x \) in the domain where both functions are defined.

Understanding composition is fundamental to exploring deeper relationships between functions and solving complex function-related problems.

Function Inverses Proof

Proving that two functions are inverses requires us to use the definition of inverse functions. The definition states that two functions \( f \) and \( g \) are inverses if the compositions \( f(g(x)) \) and \( g(f(x)) \) both equal \( x \).

• The process starts with substitution where we insert one function into the other.
• For example, if \( g(x) \) is substituted in \( f(x) \), and the result simplifies to the original input \( x \), then the composition \( f(g(x)) = x \) holds.
• Similarly, performing \( g(f(x)) \) should also result in \( x \). This confirms both functions return the input, making them inverses.
• This analytic verification demonstrates that invertible functions effectively "undo" each other when composed in either order.

This manipulation guarantees that every operation done by one function is reversed by its inverse.

Cube Root Function

Understanding the cube root function is essential when dealing with cubic transformations and solving equations related to inverse functions. The cube root function \( \sqrt[3]{x} \) is the inverse of a cubic function \( x^3 \). This means that these two functions can "undo" each other's operations.

• The cube root function helps in finding the real number that, when cubed, yields the given \( x \).
• Specifically, given a cube root \( \sqrt[3]{x} \), you can raise this to the power of three (cubing it) to obtain the original number \( x \).
• In the context of verifying inverse functions, this property assists in simplifying expressions to confirm if two functions are truly inverses.

Hence, the cube root and cubic functions are perfect examples of inverse functions because one operation precisely reverses the effect of the other.