Problem 69 - Algebra for College Students

Question

Are the functions \(f(x)=x^{4}\) and \(g(x)=\sqrt[4]{x}\) inverses of each other? Explain your answer.

Step-by-Step Solution

Answer

No, they are not inverses because \(g(f(x)) = |x| \neq x\).

Step 1: Definition of Inverse Functions

Two functions \(f(x)\) and \(g(x)\) are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in their respective domains. Let's test both conditions for these functions.

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

Start by substituting \(g(x)\) into \(f(x)\):\[ f(g(x)) = f(\sqrt[4]{x}) = (\sqrt[4]{x})^4 = x. \]So, \(f(g(x)) = x\).

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

Now substitute \(f(x)\) into \(g(x)\):\[ g(f(x)) = g(x^4) = \sqrt[4]{x^4} = |x|. \] Here, \(g(f(x)) = |x|\), not \(x\).

Step 4: Conclusion About Inverses

Since \(g(f(x)) = |x|\) and not \(x\), the functions \(f(x) = x^4\) and \(g(x) = \sqrt[4]{x}\) are not inverses over the entire domain.

Key Concepts

Domain and Range

Understanding the domain and range of functions is crucial when discussing inverse functions. **Domain** refers to all possible input values a function can accept, while **range** is all the possible output values.For the functions involved in this exercise:

• The **domain** of the function \( f(x) = x^4 \) is all real numbers \( (-\infty, \infty) \) because you can raise any real number to the fourth power.
• The **range** of \( f(x) = x^4 \) is non-negative real numbers \( [0, \infty) \) since raising any real number to an even power always results in zero or a positive result.
• The **domain** of \( g(x) = \sqrt[4]{x} \) is non-negative real numbers \( [0, \infty) \) because the fourth root of a negative number is not considered real.
• The **range** of \( g(x) = \sqrt[4]{x} \) is also non-negative real numbers \( [0, \infty) \).

Understanding these domains and ranges helps to determine why \(f(x)\) and \(g(x)\) might not be perfect inverses for all real numbers.

Function Composition

Function composition involves applying one function to the results of another, such as \(f(g(x))\) and \(g(f(x))\). To determine if two functions are inverses, these compositions need to result in the identity function, \(x\), over their domains.When we check the composition:

• For \(f(g(x))\), substituting \(g(x) = \sqrt[4]{x}\) into \(f(x)\) yields \((\sqrt[4]{x})^4 = x\), confirming this part works out perfectly.
• However, for \(g(f(x))\), substituting \(f(x) = x^4\) into \(g(x)\) gives \(\sqrt[4]{x^4} = |x|\), resulting in the absolute value of \(x\) instead of simply \(x\).

This discrepancy, where \(g(f(x)) eq x\), indicates that these functions cannot be inverses over their entire domain.

Absolute Value

The concept of absolute value emerges in the analysis of these particular functions. The **absolute value** of a number \(x\), denoted as \(|x|\), is always non-negative and represents the distance of that number from zero on the number line.In the context of this exercise:

• When \(g(f(x)) = \sqrt[4]{x^4} = |x|\) was calculated, the result was an absolute value because the power of four eliminates the sign of \(x\), always returning a non-negative result.
• This use of absolute value explains why \(g(f(x))\) does not equal \(x\) for all \(x\), especially for negative numbers.

The presence of absolute value in the function composition is a key reason \(f(x)\) and \(g(x)\) are not inverses, as it changes the expected outcome from \( \pm x \) to \( |x| \).