Problem 28 - 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

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

Step 1: Verify that applying g to f(x) returns x

Start by substituting the expression for \(f(x)\) into \(g(x)\). In mathematical terms, this is \(g(f(x))\). We have:\[g(f(x)) = g(x^3 + 1)\]Now, let's substitute this into the equation for \(g\):\[g(x^3 + 1) = ((x^3 + 1) - 1)^{\frac{1}{3}}\]Simplify the expression:\[= (x^3)^{\frac{1}{3}}\]\(= x\).This shows that \(g(f(x)) = x\).

Step 2: Verify that applying f to g(x) returns x

Next, substitute the expression for \(g(x)\) into \(f(x)\) to find \(f(g(x))\). We have:\[f(g(x)) = f((x-1)^{\frac{1}{3}})\]Now, let's substitute this into the equation for \(f\):\[f((x-1)^{\frac{1}{3}}) = (((x-1)^{\frac{1}{3}})^3 + 1)\]Simplify the powers:\[= (x-1) + 1\]\(= x\).This shows that \(f(g(x)) = x\).

Step 3: Conclusion: f and g are inverses

Since applying \(g\) to \(f(x)\) leads back to \(x\) and applying \(f\) to \(g(x)\) also returns \(x\), we show that \(f\) and \(g\) are indeed inverse functions of each other.

Key Concepts

Inverse Function Property

Inverse functions are fundamental concepts in algebra and calculus. They essentially reverse each other's effects on a variable. If you have a function \(f\) and another function \(g\), they are considered inverses if applying one function after the other returns the original input value. In mathematical terms, this means \(f(g(x)) = x\) and \(g(f(x)) = x\). This mutual cancellation property is often checked through function composition. The presence of inverse functions is useful as it means you can undo the actions of one function with the other. This is especially helpful in solving equations, simplifying expressions, or changing the focus of a problem.

Function Composition

Composition of functions refers to applying one function to the results of another. It is symbolized as \((g \circ f)(x)\) or \(g(f(x))\). The main idea is to take the output of \(f(x)\) and use it as the input of \(g(x)\).

• For the given exercise, we check both \(g(f(x))\) and \(f(g(x))\).
• These compositions are verified if both result in the original input, \(x\).

Understanding function composition is essential since it helps in detecting if functions are inverses. This relationship is crucial for verifying inverse functions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebra. It is the process that allows us to verify inverse functions step-by-step.

• In the first step of the exercise, \(g(f(x)) = x\) is found by simplifying the composed expression \(((x^3 + 1) - 1)^{1/3}\) to \(x\).
• The second step involves showing \(f(g(x)) = x\) which requires expanding \(((x-1)^{1/3})^3 + 1\) to \(x\).

If you can simplify an expression to prove the inverse property, it means you can confirm two functions are indeed inverses. This is why algebraic skills are essential in dealing with expressions and complex functions.

Cubic Functions

Cubic functions are polynomial functions where the highest degree of any variable is three. For example, in the given exercise, \(f(x) = x^3 + 1\) is a cubic function, as it includes \(x^3\) as the highest power.

• Cubic functions tend to have one real root and can have complex roots.
• They are one-to-one functions, meaning each input corresponds to one unique output.

The cubic nature of \(f(x)\) helps in its invertibility, because one-to-one functions naturally allow for the determination of an inverse. Understanding cubic functions assists in comprehending their properties, such as the role of increasing behaviors in functions, and why they possess inverses under proper conditions.