Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=x^{3}-7, \quad g(x)=\sqrt[3]{x+7}$$

Step-by-Step Solution

Answer

Yes, \( f(x) = x^3 - 7 \) and \( g(x) = \sqrt[3]{x + 7} \) are inverse functions.

Step 1: Understand the Definition of Inverse Functions

Two functions \( f \) and \( g \) are inverses if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in their respective domains. We need to verify these two conditions for the given functions.

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

Substitute \( g(x) = \sqrt[3]{x + 7} \) into \( f \): \[ f(g(x)) = f(\sqrt[3]{x+7}) = (\sqrt[3]{x+7})^3 - 7. \] Simplify the expression: \[ (\sqrt[3]{x+7})^3 = x + 7. \] Thus, \[ f(g(x)) = (x+7) - 7 = x. \] This satisfies the first condition of inverse functions.

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

Substitute \( f(x) = x^3 - 7 \) into \( g \): \[ g(f(x)) = g(x^3 - 7) = \sqrt[3]{(x^3 - 7) + 7}. \] Simplify the expression: \[ \sqrt[3]{x^3 - 7 + 7} = \sqrt[3]{x^3} = x. \] This satisfies the second condition of inverse functions.

Step 4: Conclusion

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) = x^3 - 7 \) and \( g(x) = \sqrt[3]{x + 7} \) are indeed inverses of each other.

Key Concepts

Function Composition

Function composition is a mathematical operation where the output of one function becomes the input of another. It's like a function within a function. When we say we're performing function composition, we're usually denoting it as \( f(g(x)) \). This means you take the function \( g(x) \), perform its operations, and use the result as the input for \( f \).

To check if two functions are inverses, use function composition with these criteria:

• First, calculate \( f(g(x)) \). The result should be \( x \).
• Then, do \( g(f(x)) \). Again, the result should be \( x \).

This step-by-step check ensures that each function exactly undoes the action of the other.

In our original exercise, substituting \( g(x) = \sqrt[3]{x+7} \) into \( f \) and simplifying gave us \( f(g(x)) = x \). Similarly, replacing \( f(x) = x^3-7 \) in \( g \) led to \( g(f(x)) = x \). Hence, the functions undo each other perfectly and are inverses.

Cube Roots

The concept of cube roots is fundamental when dealing with inverse functions that include cubic expressions. The cube root of a number \( y \) is a value that, when raised to the power of three, equals \( y \). In mathematical terms, the cube root of \( y \) is written as \( \sqrt[3]{y} \).

Understanding how cube roots interact with cubes is essential:

• When you cube \( \sqrt[3]{y} \), you end up with \( y \). This is because the cube root and cube operations cancel each other out.
• This interaction between cubes and cube roots is what simplifies functions like \( g(x) = \sqrt[3]{x+7} \) when composing with functions like \( f(x) = x^3 - 7 \).

For the given exercise, seeing \( \sqrt[3]{x+7}^3 = x+7 \) allows us to simplify \( f(g(x)) \) effectively. Similarly, simplifying \( \sqrt[3]{x^3} \) to \( x \) verifies \( g(f(x)) \). These simplifications account for how cube roots allow the functions \( f \) and \( g \) to be perfectly inverse of each other.

Algebraic Verification

Algebraic verification involves logically and methodically proving that two functions are inverses through clear calculation. This process underpins the overall objective of establishing the inverse relationship by ensuring every algebraic step leads to the confirmation criteria of invertibility, namely \( f(g(x)) = x \) and \( g(f(x)) = x \) for all applicable values of \( x \).

Let's break down the algebraic steps in our exercise's verification:
- You start with clear definitions and substitutions; knowing that function \( g \) substitutes into \( f \), and vice versa, is critical in this understanding.
- The simplifications, such as reducing \( (\sqrt[3]{x+7})^3 \) to \( x+7 \) and afterward subtracting 7, directly show a return to the \( x \) value.
- Similarly, simplifying the expression \( \sqrt[3]{x^3} \) leads neatly to confirming \( g(f(x)) = x \).

These algebraic checks are also an excellent rehearsal of underlying algebra rules and principles. Proper algebraic verification not only confirms the functions are inverses but also improves problem-solving skills and confidence in handling complex functions.