Problem 71 - A Graphical Approach to College Algebra

Question

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=\sqrt[3]{5 x+4}, g(x)=\frac{1}{5} x^{3}-\frac{4}{5}$$

Step-by-Step Solution

Answer

Functions \(f\) and \(g\) are inverses of each other, satisfying both \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\).

Step 1: Understand the Composition of Functions

To prove the conditions \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\), we need to start by understanding what these compositions mean. \((f \circ g)(x)\) means applying the function \(g\) first and then applying \(f\) to the result. Similarly, \((g \circ f)(x)\) means applying the function \(f\) first and then applying \(g\) to the result.

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

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). We have:\(g(x) = \frac{1}{5}x^3 - \frac{4}{5}\).Now substitute into \(f(x)\):\(f(g(x)) = f\left(\frac{1}{5}x^3 - \frac{4}{5}\right) = \sqrt[3]{5\left(\frac{1}{5}x^3 - \frac{4}{5}\right) + 4}\).Simplifying this, we get:\( = \sqrt[3]{x^3 - 4 + 4} = \sqrt[3]{x^3} = x\).This shows \((f \circ g)(x) = x\).

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

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). We have:\(f(x) = \sqrt[3]{5x + 4}\).Now substitute into \(g(x)\):\(g(f(x)) = g(\sqrt[3]{5x + 4}) = \frac{1}{5}(\sqrt[3]{5x + 4})^3 - \frac{4}{5}\).This simplifies to:\(= \frac{1}{5}(5x + 4) - \frac{4}{5} = x + \frac{4}{5} - \frac{4}{5} = x\).This proves \((g \circ f)(x) = x\).

Step 4: Conclusion

Since we have shown that \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\), the functions \(f\) and \(g\) are inverses of each other. This means any \(x\) transformed first by \(g\) and then by \(f\) (or vice versa) will result back in \(x\). The requirement set by the exercise is satisfied.

Key Concepts

Function Composition

Function composition is when you apply one function to the result of another. It can be compared to following a recipe: you take the output from the first step and use it as the input for the next. Think of two functions \( f \) and \( g \). When we say \((f \circ g)(x)\), we first apply \( g \) to \( x \), and then use this result as the input for \( f \). This idea of applying functions in sequence is fundamental because it demonstrates how outputs transform through successive operations.

In the exercise, we're dealing with the functions \( f(x) = \sqrt[3]{5x + 4} \) and \( g(x) = \frac{1}{5}x^3 - \frac{4}{5} \). When composing \( f \) and \( g \) in both possible orders, \((f \circ g)\) and \((g \circ f)\), we expect to return the input \( x \). This tells us the inputs and outputs align perfectly, making these functions special pairs known as inverses.

Algebraic Functions

Algebraic functions are created using algebraic operations such as addition, subtraction, multiplication, and division combined with raising to a power or extracting roots. These functions form the backbone of much of basic mathematical reasoning and provide simple yet powerful modeling capabilities.

In the problem at hand, our functions \( f \) and \( g \) have specific algebraic forms:

• \( f(x) = \sqrt[3]{5x + 4} \) is formed by taking a cube root.
• \( g(x) = \frac{1}{5}x^3 - \frac{4}{5} \) involves dividing and multiplying before employing subtraction.

These compositions ensure that you alter the shape and direction of \( x \)'s transformation uniquely, especially since they reverse each other's operations. Understanding their algebraic nature helps us see why they fit the conditions of inverses.

Mathematical Proof

A mathematical proof is a logical argument showing that a statement is true beyond doubt. For inverse functions, this means demonstrating two given functions perfectly undo each other through composition. In our example, we must check that \((f \circ g)(x) = x \) and \((g \circ f)(x) = x \).

To prove \((f \circ g)(x) = x\), evaluate this by substituting \( g(x) \) into \( f(x) \), giving us:

• Start with \( g(x) = \frac{1}{5}x^3 - \frac{4}{5} \).
• After substitution and simplification, you see \( f(g(x)) = x \).

Similarly, verify \((g \circ f)(x) = x\) by reversing the process:

• Start with \( f(x) = \sqrt[3]{5x + 4} \).
• Substitute into \( g \) and simplify to see \( g(f(x)) = x \).

After both compositions reduce to \( x \), we conclude the functions are inverses. Every step in this logical flow ensures there are no leaps or assumptions, just clear connections.