Problem 32 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=x^{3}+1 ; \quad g(x)=(x-1)^{1 / 3} $$

Step-by-Step Solution

Answer

Functions \(f\) and \(g\) are inverses of each other.

Step 1: Verify \(f(g(x)) = x\)

To confirm that \(f\) and \(g\) are inverses, we need to verify that performing \(f\) on \(g(x)\) results in the original \(x\). Let's compute \(f(g(x))\):Substitute \(g(x) = (x-1)^{1/3}\) into \(f(x)\): \[f(g(x)) = f((x-1)^{1/3}) = ((x-1)^{1/3})^3 + 1 \]Using the property \(((x-1)^{1/3})^3 = x-1\), we have: \[f(g(x)) = (x-1) + 1 = x\]Hence, \(f(g(x)) = x\).

Step 2: Verify \(g(f(x)) = x\)

Next, we need to confirm that applying \(g\) to \(f(x)\) also results in the original \(x\). Let's compute \(g(f(x))\):Substitute \(f(x) = x^3 + 1\) into \(g(x)\):\[g(f(x)) = g(x^3 + 1) = ((x^3 + 1) - 1)^{1/3} \]Simplify the expression: \[g(f(x)) = (x^3)^{1/3} = x\]Thus, \(g(f(x)) = x\).

Key Concepts

Cubic Functions

Cubic functions are polynomial functions where the highest degree of the variable is three. This makes them essential in many mathematical situations due to their unique properties. A general cubic function is expressed as \(f(x) = ax^3 + bx^2 + cx + d\), where \(a \, eq \, 0\).
They have the following characteristics:

• Graph Shape: Cubic functions produce an "S" shaped curve on a graph, known as a cubic curve or a cubic parabola. It can have one, two, or three real roots.
• Endpoints: As \(x\) approaches positive or negative infinity, the cubic function's value also heads toward infinity or negative infinity, depending on the leading coefficient \(a\).
• Symmetry: Although not symmetric like quadratic functions, cubic functions can have rotational symmetry depending on the coefficients.

Understanding cubic functions is important in the study of inverses, as the inverse of a cubic function can often be expressed in terms of cube roots. This relationship is key when verifying inverses, as seen with the functions \(f(x) = x^3 + 1\) and \(g(x) = (x-1)^{1/3}\).

Composition of Functions

Composition of functions involves applying one function to the results of another function. Understanding this concept is crucial when working with inverse functions because it allows us to check if two functions are indeed inverses of each other.
If you have two functions, \(f\) and \(g\), the composition \(f(g(x))\) means you first apply \(g\) to \(x\), and then apply \(f\) to the result:

• The notation \(f(g(x))\) can be read as "\(f\) of \(g\) of \(x\)."
• This process can often be simplified using algebraic manipulation, as seen when simplifying \(f(g(x)) = ((x-1)^{1/3})^3 + 1\) to \(x\).
• Using composition to check inverses requires showing that both \(f(g(x)) = x\) and \(g(f(x)) = x\).

Composition of functions may seem complex at first, but with practice, it becomes a powerful tool for solving diverse mathematical problems, particularly those involving inverse functions.

Function Properties

Function properties are rules and behaviors that functions follow. Knowing these properties helps in manipulating and understanding functions, particularly when establishing that two functions are inverses. Here are some key function properties related to inverses:

• Inverse Function Property: For two functions, \(f\) and \(g\), to be inverses, they must satisfy \(f(g(x)) = x\) and \(g(f(x)) = x\). This means applying one function after the other returns the original input.
• Bijective Functions: A function needs to be one-to-one (injective) and onto (surjective) to have an inverse. This ensures each input has a unique output and covers all possible outputs.
• Notation and Symbols: The inverse of a function \(f\) is often denoted as \(f^{-1}\). However, remember this does not mean \(1/f\). It represents a new function that undoes \(f\).

These properties help to understand how and why we're able to find and confirm that functions are inverses of each other. By verifying that the compositions \(f(g(x))\) and \(g(f(x))\) both equal \(x\), we see these properties in action.