Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=-x^{5}, \quad g(x)=-\sqrt[5]{x}$$

Step-by-Step Solution

Answer

Functions \(f(x) = -x^5\) and \(g(x) = -\sqrt[5]{x}\) are inverse functions.

Step 1: Understand the Concept of Inverse Functions

Two functions \(f\) and \(g\) are considered inverse functions if the function \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domains of the respective compositions. We will check both conditions to determine if \(f\) and \(g\) are inverses.

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

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). We have:\[g(x) = -\sqrt[5]{x}\]So, substituting into \(f(x)\):\[f(g(x)) = f(-\sqrt[5]{x}) = -(-\sqrt[5]{x})^{5}\]Simplifying this, we get:\[-(-1 \cdot x^{1/5})^{5} = -(-1)^{5} \cdot (x^{1/5})^{5} = -(-1) \cdot x = x\]Thus, \(f(g(x)) = x\).

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

To find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). We have:\[f(x) = -x^{5}\]So, substituting into \(g(x)\):\[g(f(x)) = g(-x^{5}) = -\sqrt[5]{-x^{5}}\]Simplifying this, we get:\[-(-1 \cdot x^{5})^{1/5} = -(-1) \cdot x^{5 \cdot 1/5} = -(-1) \cdot x = x\]Thus, \(g(f(x)) = x\).

Step 4: Conclusion

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\), we have shown that \(f\) and \(g\) are indeed inverse functions by definition.

Key Concepts

Composition of Functions

When dealing with inverse functions, the idea of function composition plays a crucial role. To say that two functions are inverses of each other, we must use their compositions. Function composition involves plugging one function into another and is written as either \(f(g(x))\) or \(g(f(x))\).

• The composition \(f(g(x))\) means you take the output of \(g(x)\) and use it as the input for \(f(x)\).
• The composition \(g(f(x))\) means you take the output of \(f(x)\) and use it as the input for \(g(x)\).

To determine if two functions, \(f\) and \(g\), are inverses, we calculate both \(f(g(x))\) and \(g(f(x))\). If both results simplify to \(x\), the functions undo each other's operations, proving they are inverses. It is like traveling a path, then tracing your steps back exactly to your starting point. Thus, showing \(f(g(x)) = x\) and \(g(f(x)) = x\) analytically can be compared to proving you can go there and back again without any deviation.

Domain of Functions

Understanding the domain of functions is vital, especially when dealing with compositions and inverses. The domain of a function is the set of all possible input values (\(x\)) that will output valid results when passed through the function. In composition of inverse functions, the domains must align correctly to ensure smooth operations. The domain of \(f(g(x))\) must be considered as the domain of \(g(x)\) plus any restrictions imported from \(f(x)\). Similarly for \(g(f(x))\), the domain depends on \(f(x)\) with any additional limitations imposed by \(g(x)\). In our current problem:

• For \(f(x)=-x^5\), the domain is all real numbers \((-\infty, \infty)\) because any real number can be raised to the fifth power.
• For \(g(x)=-\sqrt[5]{x}\), the domain is also all real numbers since the fifth root of any real number exists.

Ensuring correct domain consideration means checking compositions stuff makes logical mathematical sense. Practically, it ensures we don't perform operations on nonexistent or undefined inputs. Proper domain recognition is essential to validate function inverses since the operations involved must work soundly for all relevant \(x\).

Algebraic Simplification

Algebraic simplification is the process of reducing expressions making them easier to handle and understand. It is especially helpful when verifying inverse functions as seen here, where simplification lets us see if \(f(g(x))\) or \(g(f(x))\) return the identity element \(x\).To simplify expression like \(f(g(x)) = -(-(\sqrt[5]{x}))^5\), the goal is to break down the nested functions using algebraic rules:- Apply the exponent \((...)^5\) properly ensuring that both negative signs and roots appear correctly.- When simplifying \(-(-1 \cdot x^{1/5})^5\), calculate stepwise moving from inside out. The negative sign multiplied by itself 5 times yields another negative sign. Then you proceed to raise \(x^{1/5}\) to the fifth power resulting again in \(x\).The process is similar for \(g(f(x)) = -\sqrt[5]{-x^5}\). Here:

• The negative cube root neutralizes internal negatives, keeping aware of odd-even interactions.
• Expressing \(-(-1 \cdot x^{5})^{1/5}\), this alve again encapsulates value shifts cancelling appropriately.

Simplifying until the original input \(x\) reappears demonstrates the simplification shows inverse relationships clearly connecting expressed values with algebraic grounding.