Problem 32 - Contemporary Precalculus

Question

Verify that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\) for every \(x\) $$f(x)=2 x^{3}-5, \quad g(x)=\sqrt[3]{\frac{x+5}{2}}$$

Step-by-Step Solution

Answer

Answer: Yes, the functions \(f(x)=2x^{3}-5\) and \(g(x)=\sqrt[3]{\frac{x+5}{2}}\) are inverse functions of each other for every \(x\).

Step 1: Find \((f \circ g)(x)\)

To find the composition \((f \circ g)(x)\), we need to replace the x in the function f(x) with the expression for g(x), like this: \(f(g(x))\). Let's substitute g(x) into f(x): $$ f(g(x))=2(\sqrt[3]{\frac{x+5}{2}})^{3}-5 $$Now simplify the expression:$$ f(g(x))=2\left(\frac{x+5}{2}\right)-5 $$And finally:$$ f(g(x))=x $$We can see that \((f \circ g)(x)=x\).

Step 2: Find \((g \circ f)(x)\)

To find the composition \((g \circ f)(x)\), we need to replace the x in the function g(x) with the expression for f(x), like this: \(g(f(x))\). Let's substitute f(x) into g(x): $$ g(f(x))= \sqrt[3]{\frac{2 x^{3}-5+5}{2}} $$Now simplify the expression:$$ g(f(x))=\sqrt[3]{\frac{2 x^{3}}{2}} $$And finally:$$ g(f(x))=x $$We can see that \((g \circ f)(x)=x\).

Step 3: Conclusion

Since both \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\), we have verified that the given functions f(x) and g(x) are indeed inverse functions of each other for every \(x\).

Key Concepts

Composition of Functions

The composition of functions involves combining two functions in such a way that the output of one function becomes the input for another. In terms of notation, if we have two functions, \(f(x)\) and \(g(x)\), their composition is noted as \((f \circ g)(x)\) or simply \(f(g(x))\). This means we take \(g(x)\) first and plug its result into \(f(x)\).

• In the exercise, we composed both \(f(x)\) and \(g(x)\) to verify inverse functions.
• We did this by calculating \((f \circ g)(x)\) and \((g \circ f)(x)\), both resulting in \(x\), demonstrating they undo each other.

When these function compositions return the same result \(x\), it supports the fact that the functions are inverses.

Function Notation

Function notation is a simple yet powerful way of representing functions and their operations. Instead of writing lengthy expressions every time, we denote them by symbols such as \(f(x)\), where \(f\) represents the function and \(x\) is the input variable. This notation helps to easily manipulate and transform functions when performing operations such as composition.

• In the exercise example, the notation \(f(x) = 2x^3 - 5\) helped us understand that this function accepts an input \(x\), cubes it, doubles it, and then subtracts five.
• Similarly, \(g(x) = \sqrt[3]{\frac{x+5}{2}}\) denotes a function applying the cube root to a transformed \(x\).

Using this consistent notation allows us to seamlessly substitute and compose functions, as demonstrated in verifying the inverses.

Cube Root

The cube root is a specific type of root that finds the number which, when multiplied by itself three times, results in the original number. In symbols, \(\sqrt[3]{x}\) is the cube root of \(x\). Cube roots have unique properties:

• Cube roots can be applied to both positive and negative numbers.
• The operation of cubing a cube root effectively cancels it out, resulting in the original number, embodied by \((\sqrt[3]{x})^3 = x\).

In the context of functions, as observed in \(g(x) = \sqrt[3]{\frac{x+5}{2}}\), the cube root is instrumental in reversing cubic transformations of \(f(x)\), thereby facilitating the verification of inverse relationships when \((f \circ g)(x)\) and \((g \circ f)(x)\) both simplified to \(x\).

Polynomial Functions

Polynomial functions encompass expressions comprised of variables raised to whole-number exponents, each multiplied by coefficients, typically written in descending order of the degree. The function \(f(x) = 2x^3 - 5\) is a polynomial of degree 3 because the highest power of \(x\) is 3.

• Polynomials of degree 3 are called cubic polynomials, capable of having up to three real roots.
• Such functions can be inverted using appropriate techniques leading to transformations like those applied to find the inverse using \(g(x)\).

Understanding both polynomials and their inverse operations helps in tackling compositions and solving equations that deal with powers and roots, as was crucial to verifying the inverses of \(f(x)\) and \(g(x)\).