Problem 48 - Precalculus Essentials

Question

a. Find an equation for \(f^{-1}(x)\) b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\) $$f(x)=(x-2)^{3}$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{x}+2\). Both functions \(f\) and \(f^{-1}\) can be graphed on the same Cartesian plane with their graphs being reflections of each other. The domain and range for both the functions \(f\) and \(f^{-1}\) are \(-\infty\) to \(+\infty\).

Step 1: Find the Inverse Function

To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\) for simplicity. So, \(y = (x-2)^3\). Switch \(x\) and \(y\) to get \(x = (y-2)^3\). Solve this equation for \(y\). We get \(f^{-1}(x) = \sqrt[3]{x}+2\). This is the inverse function of \(f\).

Step 2: Graph the Functions

To graph \(f\) and \(f^{-1}\), plot some points for each function and connect them properly. For function \(f\), important points are (0,-8), (1,-1), (2,0), (3,1), and (4,8). For the inverse function \(f^{-1}\), important points are (-8,0), (-1,1), (0,2), (1,3), and (8,4). Connect the respective points with curved lines. Notice that \(f^{-1}\) is the reflection of \(f\) across the line \(y=x\).

Step 3: Find the Domain and Range

The domain of a function is the set of all allowable input values (x-values), while the range is the set of possible output values (y-values). For function \(f\), since any real number can be plugged into the equation and a real number result obtained, the domain is \(-\infty\) to \(+\infty\). The range, based on the graph, is also \(-\infty\) to \(+\infty\). The inverse function \(f^{-1}\) will have the domain and range swapped with function f, so it's also \(-\infty\) to \(+\infty\).

Key Concepts

Graphing Functions

Graphing a function visually represents its behavior and is essential for understanding its characteristics. To graph a function like \(f(x) = (x-2)^3\), you start by selecting a set of input values (\(x\)-values). For each input, calculate the corresponding output (\(y\)-value). By plotting the key points and considering the continuous nature of polynomials, you can sketch the function's curve.

When graphing its inverse, \(f^{-1}(x) = \sqrt[3]{x} + 2\), the process is similar, but with a twist. The graph of the inverse is a reflection of the original function across the line y=x. Therefore, when you plot \(f^{-1}(x)\)'s points, you'll notice that for each point (a, b) on \(f(x)\), there is a corresponding point (b, a) on \(f^{-1}(x)\).

• To graph \(f(x)\), you could plot points such as (0, -8), (1, -1), (2, 0), (3, 1), and (4, 8).
• For \(f^{-1}(x)\), the points would be (-8, 0), (-1, 1), (0, 2), (1, 3), and (8, 4).

By smoothly connecting these points, you create an accurate depiction of the functions and their inverses.

Domain and Range

The domain of a function is the complete set of possible values of the independent variable, or inputs, which in our case are \(x\)-values. For the cubic function \(f(x) = (x-2)^3\), since cubes exist for all real numbers, the domain is all real numbers, using interval notation, we write this as \( (-\infty, \infty) \).

The range of a function represents the complete set of possible output values, which are \(y\)-values. For our example function, \(f(x)\), this is also all real numbers, since cubic functions can produce any real value given the appropriate input. So, the range is also \( (-\infty, \infty) \).

Understanding the domain and range is important because it tells us what the function can 'accept' as inputs (domain) and what it can 'produce' as outputs (range). Each function has its own domain and range, which are essential aspects to comprehend when analyzing function behavior.

Interval Notation

Interval notation is a shorthand way to express the domain and range of functions. It uses parentheses and brackets to describe intervals of continuous values. In interval notation, parentheses, '(', ')', indicate that the endpoints are not included, known as open intervals, while brackets, '[', ']', indicate that endpoints are included, called closed intervals.

For example, the domain of \(f(x)\) which consists of all real numbers is written as \( (-\infty, \infty) \). Since \(-\infty\) and \(\infty\) are not real numbers and cannot be reached, we use parentheses. If the domain were just numbers greater than or equal to 0, you would write this as \( [0, \infty) \).

Interval notation is both concise and precise; thus, it's favored for expressing the set of possible values for functions. When working with functions and their inverses, understanding interval notation is crucial for effectively communicating domains and ranges.