Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Let \(n\) be any positive integer. Find the inverse function of \(f\) if (a) \(f(x)=x^{n}\) for \(x \geq 0\) (b) \(f(x)=x^{m / n}\) for \(x \geq 0\) and \(m\) any positive integer

Step-by-Step Solution

Answer

(a) \( f^{-1}(x) = x^{1/n} \); (b) \( f^{-1}(x) = x^{n/m} \).

Step 1: Understand the inverse function

To find an inverse function, we need to find a function that will undo the effect of the original function. If \( y = f(x) \), the inverse function \( f^{-1}(y) \) gives back \( x \).

Step 2a: Solve for the inverse of \( f(x) = x^n \)

Given \( y = x^n \), to find \( x \) in terms of \( y \), we solve \( x = y^{1/n} \) because raising \( y \) to the power \( 1/n \) will undo the \( n \)-th power. Thus, the inverse function is \( f^{-1}(x) = x^{1/n} \).

Step 3: Verify the inverse for \( f(x) = x^n \)

To ensure we have the correct inverse, check that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). For \( f(f^{-1}(x)) = (x^{1/n})^n = x \) and \( f^{-1}(f(x)) = (x^n)^{1/n} = x \) confirms they are inverses.

Step 2b: Solve for the inverse of \( f(x) = x^{m/n} \)

Given \( y = x^{m/n} \), solve for \( x \) by raising both sides to the power \( n/m \): \( x = y^{n/m} \). This inverses the effect of the \( m/n \)-th power. Therefore, the inverse function is \( f^{-1}(x) = x^{n/m} \).

Step 4: Verify the inverse for \( f(x) = x^{m/n} \)

To verify, check \( f(f^{-1}(x)) = (x^{n/m})^{m/n} = x \) and \( f^{-1}(f(x)) = (x^{m/n})^{n/m} = x \). Both identities confirm the functions are inverses.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number, called the base, is raised to the power of an exponent. The exponent tells us how many times the base is multiplied by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, meaning \(2 \times 2 \times 2 = 8\).

When the exponent is a positive integer, it is straightforward: multiply the base by itself as many times as indicated by the exponent.

• Example: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)

This operation is essential to understanding inverse functions, as working with exponents often involves reversing the process, a key step in finding inverse functions. Hence, understanding exponentiation is crucial in mathematics, especially when dealing with function transformations and solving equations.

Positive Integers

Positive integers are all the whole numbers greater than zero. They form the set \( \{1, 2, 3, 4, \ldots \} \). Positive integers are fundamental in operations such as exponentiation, where they often act as the exponents.

In functions like \(f(x) = x^n\), \(n\) is a positive integer, indicating that the base \(x\) is raised to the \(n\)-th power. This understanding allows us to reverse the function by engaging with inverse operations, such as raising a number to the fractional power \(1/n\). Positive integers are simple yet powerful numbers in mathematics that allow for diverse operations and form the backbone of arithmetic and algebraic functions.

Verification of Inverse

Verifying that a function is the inverse of another involves a two-step check that guarantees the operations reverse each other. If you find the inverse function \(f^{-1}(x)\) of \(f(x)\), then plugging \(f^{-1}\) into \(f\) should return you to the original input \(x\). Likewise, placing \(f\) into \(f^{-1}\) should also return \(x\).

To illustrate, if \(f(x) = x^n\) and its inverse \(f^{-1}(x) = x^{1/n}\), then:

• Checking \(f(f^{-1}(x))\): \((x^{1/n})^n = x\)
• Checking \(f^{-1}(f(x))\): \((x^n)^{1/n} = x\)

This process ensures that both operations are indeed inverses, as applying one undoes the effect of the other. Verification is a vital step to confirm that we have accurately determined the inverse function, maintaining integrity in mathematical problem-solving.