Problem 52 - Precalculus with Limits

Question

(a) find the inverse function of \(f\) (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). \(f{x} = x^6 + 1\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x^6 + 1\) is \(f^{-1}(x) = \sqrt[6]{x - 1}\). The graph of \(f^{-1}\) is a reflection of the graph of \(f\) over the line \(y = x\). The domain and range of \(f\) are all real numbers and \(y > 1\) respectively, while the domain and range of \(f^{-1}\) are \(x > 1\) and all real numbers respectively.

Step 1: Find the Inverse Function of \(f\)

To find the inverse function, \(f^{-1}(x)\), of a given function, \(f(x) = x^6 + 1\), first replace \(f(x)\) with \(y\): \(y = x^6 +1\). Then swap \(x\) with \(y\): \(x = y^6 + 1\). Finally, solve the equation for \(y\) to obtain the inverse function: \(f^{-1}(x) = \sqrt[6]{x - 1}\). Note that the real sixth root is used because we're dealing with real numbers in this case.

Step 2: Graph Both \(f\) and \(f^{-1}\) on The Same Set of Coordinate Axes

For the graph of function \(f(x) = x^6 + 1\), it starts from 1 (as the smallest y-value is 1) and stretches to the top of the y-axis. For the graph of its inverse function \(f^{-1}(x) = \sqrt[6]{x - 1}\), it starts from 1 (as the smallest x-value is 1) and stretches to the end of the x-axis. Then there is the line \(y = x\) which is the reflection line of a function and its inverse.

Step 3: Describe the Relationship Between the Graphs of \(f\) and \(f^{-1}\)

The graph of \(f^{-1}\) is a reflection of the graph of \(f\) over the line \(y = x\). This is always the case with a function and its inverse.

Step 4: State The Domain and Range of \(f\) and \(f^{-1}\)

For the function \(f\): The domain of \(f\) is all real numbers, as you can input any real number and the function will output a real number. The range of \(f\) is \(y > 1\), because \(x^6 + 1\) will always output a value greater than 1. For the inverse function \(f^{-1}\): The domain of \(f^{-1}\) is \(x > 1\), because you can only take the sixth root of \(x - 1\) if \(x > 1\). The range of \(f^{-1}\) is all real numbers, as any real number could be the output of the inverse function.

Key Concepts

Domain and Range

When exploring functions and their inverses, understanding the domain and range is essential. For the given function, \(f(x) = x^6 + 1\), its domain includes all real numbers. This is because any real number can be used as an input to find the corresponding output. The function will always yield a real number output, which makes its domain all-encompassing within real numbers.

The range of this function is a bit more specific. Since you are adding 1 to \(x^6\), and raising a number to the sixth power yields a non-negative result (greater than zero), the smallest outcome \(f(x)\) can yield is 1. Thus, the range of \(f\) is:

• \( y > 1 \)

This means \(f(x)\) never dips below 1.

Now, considering the inverse function, \(f^{-1}(x) = \sqrt[6]{x - 1}\), the situation flips. The domain of \(f^{-1}\) requires that \(x\) must be greater than 1 (\(x > 1\)) to ensure the sixth root remains valid and real. Thus, the possible inputs are limited. However, this makes the range of the inverse function more inclusive, covering all real numbers. Hence, even though \(f^{-1}(x)\) has a limited starting point, it can eventually reach any real number.

Function Graph Reflection

Function graph reflection is an intriguing aspect of inverse functions. When a function and its inverse are graphed on a coordinate system, a fascinating relationship is observed. Specifically, with a function \(f\) and its inverse \(f^{-1}\), the graph of \(f^{-1}\) serves as a mirror image of \(f\) across the line \(y=x\). This line is the pivot or the line of symmetry.

This reflection tells a fundamental truth about how inverse relationships between functions and their graphs work:

• Points on the graph of \(f(x)\) and \(f^{-1}(x)\) swap roles; an output of one becomes an input for the other.
• The point \((a, b)\) on the graph of \(f(x)\) will reflect to point \((b, a)\) on \(f^{-1}(x)\).

This reflective quality helps visually confirm the harmonic relationship between a function and its inverse, verifying calculations through a visual check.

Sixth Root

The sixth root plays a key role when dealing with inverse functions like \(f^{-1}(x) = \sqrt[6]{x - 1}\). Understanding the concept of roots, especially higher-order roots like the sixth, is critical.

Higher roots such as the sixth root are determined by solving for a number that, when raised to the sixth power, equals a given value. For example, finding \(\sqrt[6]{y}\) results in a value that, if multiplied by itself six times, equals \(y\). It's like asking what base number, when raised to the sixth power, yields a specific number.

• This concept is crucial in solving inverse functions, particularly for equations involving even powers, like \(x^6\).
• It opens doors to many domains, such as problem-solving in geometry and physics, where one needs to "undo" the process of exponentiation to a degree of 6.

When working with higher roots, particularly in real-valued functions, ensure inputs are adjusted so the operations remain valid, like ensuring \(x>1\) in the inverse, \(f^{-1}(x) = \sqrt[6]{x - 1}\). This preserves the reality and applicability of root operations, allowing seamless function manipulation.