Problem 58 - Intermediate Algebra

Question

Show that each pair of functions are inverses. $$ f(x)=\sqrt[3]{x-6}, f^{-1}(x)=x^{3}+6 $$

Step-by-Step Solution

Answer

Both conditions are satisfied; the functions are inverses.

Step 1: Understand the Definition of Inverse Functions

Two functions \( f \) and \( f^{-1} \) are inverses if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This means that applying one function and then its inverse should return the original input.

Step 2: Verify \( f(f^{-1}(x)) = x \)

We start by substituting \( f^{-1}(x) = x^3 + 6 \) into \( f(x) = \sqrt[3]{x - 6} \). Calculate \( f(f^{-1}(x)) = f(x^3 + 6) = \sqrt[3]{(x^3 + 6) - 6} = \sqrt[3]{x^3} = x \). This confirms that the first condition is satisfied.

Step 3: Verify \( f^{-1}(f(x)) = x \)

Now substitute \( f(x) = \sqrt[3]{x - 6} \) into \( f^{-1}(x) = x^3 + 6 \). Compute \( f^{-1}(f(x)) = f^{-1}(\sqrt[3]{x - 6}) = (\sqrt[3]{x - 6})^3 + 6 = x - 6 + 6 = x \). This confirms that the second condition is satisfied.

Step 4: Conclude Inverses are Verified

Since both conditions \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) are satisfied, the functions \( f(x) = \sqrt[3]{x - 6} \) and \( f^{-1}(x) = x^3 + 6 \) are indeed inverses of each other.

Key Concepts

Exploring Function Composition

Function composition is a fundamental idea in mathematics, involving combining two functions such that the output of one function becomes the input for another. To visualize, when you have a function \( f(x) \) and another function \( g(x) \), the composition \( f(g(x)) \) is essentially feeding the result of \( g(x) \) into \( f(x) \). This concept is applied heavily in verifying inverse functions.In the case of inverse functions, if you have \( f(x) \) and its inverse \( f^{-1}(x) \), function composition helps us check whether they are true inverses.

• For a pair of functions to be inverses, it must hold that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
• Applying a function and its inverse should lead back to the original value.

Understanding this process is crucial for working with different types of functions, particularly when confirming if two functions are indeed inverses of each other.

Understanding Cubic Functions

Cubic functions are polynomials of degree three, expressed generally as \( ax^3 + bx^2 + cx + d \). These functions have one or three real roots, depending on the discriminant, and can exhibit various shapes in their graphs, such as having an inflection point. In the context of inverse relationships, cubic functions play a vital role. For instance, if \( f^{-1}(x) \) is given as \( x^3 + 6 \), it represents a simple cubic function and offers an important perspective on how functions increase more sharply compared to linear or quadratic functions.

• Cubic functions can produce different slopes at different points, reflecting their more complex behavior.
• They come in handy particularly because their roots can take on any real value, allowing them to undo cube root transformations.

This characteristic of cubic functions makes them perfect candidates for inverse actions, especially when paired with cube root functions.

Explaining Cube Root Functions

Cube root functions are the inverse operations of cubic functions, typically expressed as \( \sqrt[3]{x} \). Unlike square roots, cube roots can accept negative numbers because cubing negative numbers results in negative outputs, which makes cube roots quite versatile. The function \( f(x) = \sqrt[3]{x - 6} \) represents a transformation of the basic cube root function, involving a shift six units to the right. This shift ensures the expression inside the root, \( x-6 \), can be negative, zero, or positive.

• Cube root functions have a continuous curve that spans all real numbers.
• They are particularly important for converting back to original inputs after a cubic transformation.

In inverse relationships, they manipulate inputs such that applying a subsequent cubic function will restore the input's original value, as demonstrated by the relationship \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).