Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=x^{7}$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x^{7}\) is \(f^{-1}(x) = x^{1/7}\). The properties \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) are verified as both equalities hold.

Step 1: Switch x and y

The first step in finding an inverse is to replace \(f(x)\) with \(y\), then switch the roles of \(x\) and \(y\). This gives the equation \(x = y^{7}\).

Step 2: Solve for y

Solving for \(y\) will give the inverse function. In this case, take the seventh root of both sides to find \(y\). The equation becomes \(y = x^{\frac{1}{7}}\).

Step 3: Write the Inverse Function

Now write \(f^{-1}(x) = x^{\frac{1}{7}}\). This means that the inverse function of \(f(x) = x^7\) is \(f^{-1}(x) = x^{\frac{1}{7}}\). This operation reverses the function \(f(x)\).

Step 4: Verify \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x) = x^{\frac{1}{7}}\) into \(f(x)\). This gives \(f(f^{-1}(x)) = (x^{\frac{1}{7}})^7 = x\). This is the identity function for \(x\), which verifies this property of inverse functions.

Step 5: Verify \(f^{-1}(f(x)) = x\)

Substitute \(f(x) = x^{7}\) into \(f^{-1}(x)\). This gives \(f^{-1}(f(x)) = (x^{7})^{\frac{1}{7}} = x\). This is also the identity function for \(x\), which verifies this property of inverse functions.

Key Concepts

Inverse Operations

Inverse operations are essential in mathematics as they allow us to reverse the effects of a function. For example, if a function multiplies a number by 7, its inverse will divide the number by 7, essentially undoing the original operation.
When working with inverse functions such as in our exercise where we have the original function as \( f(x) = x^7 \), finding the inverse involves reversing this operation. We do this by following a series of steps that eventually let us express the inverse function, in this case, as \( f^{-1}(x) = x^{\frac{1}{7}} \).
This inverse operation effectively undoes the effect of raising a number to the seventh power by taking the seventh root, thereby returning us to our original value. Understanding inverse operations is a critical skill when dealing with any mathematical functions, especially in calculus and algebra.

Identity Function

The identity function plays a crucial role in the context of inverse functions. This function is simply \( f(x) = x \), meaning that for any input, the output is the exact same value. It's a basic yet powerful concept because in the inverse verification process, it demonstrates that applying a function and its inverse results in the original identity function.
In our example, when we compute \( f(f^{-1}(x)) \), we find that it equals \((x^{\frac{1}{7}})^7 = x\). Similarly, computing \( f^{-1}(f(x)) \) results in \((x^7)^{\frac{1}{7}} = x\). Both outcomes mirror the identity function. Such equalities confirm the correct relationship between a function and its inverse, ensuring that the transformation and reversion bring us back to our original starting point \(x\).

Function Verification

In mathematics, ensuring that a function's inverse is correctly determined involves a process called function verification. This process confirms that applying a function and then its inverse should return the initial input value—demonstrating their roles as perfect opposites.
In verifying the inverse function, there are key steps: substituting \( f^{-1}(x) \) into \( f(x) \) or vice versa and ensuring the result is the identity function \( x \). This step-by-step validation was shown in the exercise, where both \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) resulted in \( x \). These results ensure mathematical accuracy and are crucial in many real-world applications, including scientific calculations and computer algorithms.

• Substitute the inverse function into the original function and check that the output is \( x \).
• Substitute the original function into the inverse and ensure the same result: \( x \).
• Observe if both verifications lead to the identity function—a clear indication of correctness.

Understanding function verification strengthens your grasp of mathematical functions and their inverses, ensuring you can apply these concepts accurately in diverse scenarios.