Problem 39 - A Graphical Approach to College Algebra

Question

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=x^{3}+4, \quad g(x)=\sqrt[3]{x-4}$$

Step-by-Step Solution

Answer

The functions \(f(x) = x^3 + 4\) and \(g(x) = \sqrt[3]{x-4}\) are inverses because both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\).

Step 1: Understand the Definition of Inverse Functions

Two functions, \(f\) and \(g\), are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domain of \(g\) and \(f\) respectively. We need to verify these conditions for the given functions.

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

Substitute the expression for \(g(x)\) into \(f(x)\). That is, substitute \(g(x) = \sqrt[3]{x-4}\) into \(f(x) = x^3 + 4\). Calculate \(f(g(x)) = \left(\sqrt[3]{x-4}\right)^3 + 4 = (x-4) + 4 = x\). This confirms \(f(g(x)) = x\).

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

Substitute the expression for \(f(x)\) into \(g(x)\). That is, substitute \(f(x) = x^3 + 4\) into \(g(x) = \sqrt[3]{x-4}\). Calculate \(g(f(x)) = \sqrt[3]{(x^3 + 4) - 4} = \sqrt[3]{x^3} = x\). This confirms \(g(f(x)) = x\).

Step 4: Confirm Both Properties of Inverse Functions

Since we have shown that both \(f(g(x)) = x\) and \(g(f(x)) = x\), by definition, the functions \(f(x) = x^3 + 4\) and \(g(x) = \sqrt[3]{x-4}\) are indeed inverse functions of each other.

Key Concepts

Understanding the Definition of Inverse Functions

Inverse functions are a fascinating concept in mathematics, often symbolized by the function notation \(f^{-1}(x)\). Two functions, \(f\) and \(g\), are considered inverses if they "undo" each other. This means that applying \(f\) followed by \(g\) returns our original input \(x\), and vice versa. Let's explore this definition more deeply:

• Inverse Function Relation: For \(f\) and \(g\) to be inverses, both \(f(g(x)) = x\) and \(g(f(x)) = x\) must hold true.
• Application in the Real World: Think of inverse functions like following directions with a map and then retracing your steps; reaching back to point A from point B.

So, when faced with a pair of functions, our goal is to show these conditions are satisfied. Confirming them mathematically, ensures the functions undo one another, thereby qualifying as inverse functions.

Exploring the Domain of Functions

The domain of a function is the complete set of possible values of the independent variable, typically \(x\). Understanding the domain is crucial when dealing with inverse functions, as each must be within the other's domain for the inverse relationship to be valid.

• Function \(f(x) = x^3 + 4\): The domain here covers all real numbers because a cubic function can take any real value.
• Function \(g(x) = \sqrt[3]{x-4}\): Likewise, a cube root function also accommodates all real numbers.

However, check that the resulting values of both functions lie within each other's domains. Since no restrictions exist in our example, this requirement is naturally satisfied. But, always verify domain constraints, especially if involved with radicals or fractions.

Analytical Verification of Inverse Functions

Analytical verification requires verifying that both \(f(g(x)) = x\) and \(g(f(x)) = x\). This process confirms the pair of functions work as perfect inverses. Here's how it applies to our pair:

• For \(f(g(x))\):
  Substitute \(g(x) = \sqrt[3]{x-4}\) into \(f(x) = x^3 + 4\):
  \[f(g(x)) = \left(\sqrt[3]{x-4}\right)^3 + 4 = (x - 4) + 4 = x\]
• For \(g(f(x))\):
  Substitute \(f(x) = x^3 + 4\) into \(g(x) = \sqrt[3]{x-4}\):
  \[g(f(x)) = \sqrt[3]{(x^3 + 4) - 4} = \sqrt[3]{x^3} = x\]

Successfully showing both these identities proves that \(f\) and \(g\) are indeed inverse functions. It's this step-by-step substitution and simplification that confirm the inverse property analytically, giving us factual proof rather than assuming based on function forms.