Problem 46 - 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 \(f^{-1}\) $$f(x)=x^{3}+1$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = x^{3} + 1\) is \(f^{-1}(x) = \sqrt[3]{x-1}\). Both functions have their domains and ranges as \((-\infty, \infty)\). The graphs of the functions show that \(f^{-1}(x)\) is a reflection of \(f(x)\) about the line \(y=x\), indicating that they are indeed inverses of each other.

Step 1: Find the inverse function

To find the inverse \(f^{-1}(x)\) of the function \(f(x)\), first replace \(f(x)\) with \(y\). This makes the equation \(y = x^{3} + 1\). Next, swap \(x\) and \(y\) and solve for \(y\). The equation becomes \(x = y^{3} + 1\). After subtracting 1 from both sides and taking the cubic root, the result is \(f^{-1}(x) = \sqrt[3]{x-1}\)

Step 2: Graphing the function and its inverse

To graph the function \(f(x)\) and its inverse \(f^{-1}(x)\), a 2D graph can be used. The function \(f(x)=x^{3}+1\) is a cubic function shifting one unit upward. Its graph is a curve starting from the point (-1, 0). The inverse function \(f^{-1}(x) = \sqrt[3]{x-1}\), is the reflection of the cubic function about the line \(y=x\), also shifted one unit to the right.

Step 3: Determine the domain and range

The domain of a function is the set of all permissible input values (x-values), while the range is the set of resultant output (y-values). For \(f(x) = x^{3}+1\), the domain and range are all real numbers, so the domain and range are \((-\infty, \infty)\). The inverse function \(f^{-1}(x) = \sqrt[3]{x-1}\) also has the domain and range as all real numbers, so its domain and range are also \((-\infty, \infty)\).

Key Concepts

Understanding Domain and Range

When dealing with functions, it's crucial to grasp the ideas of domain and range. The **domain** of a function is all the possible x-values (inputs) that will give valid outputs in the function. These are the numbers you can plug into the function without causing any mathematical issues, like division by zero or taking even roots of negative numbers.

For our function,

• The domain of \( f(x) = x^3 + 1 \) includes all real numbers. That's because you can cube any real number and then add 1 without running into any problems.

Similarly, the **range** of a function is all the possible y-values (outputs). When you plug all real numbers into \( f(x) = x^3 + 1 \), you can get any real number as a result:

• Hence, the range of \( f(x) \) is also all real numbers, or \((-finity, \infty)\).

The inverse function \( f^{-1}(x) = \sqrt[3]{x-1} \) is also defined for all real x values, and since it's essentially the reverse process, its range is also all real numbers. Understanding these concepts helps in identifying how functions behave and what kind of outputs to expect.

Graphing Functions and Their Inverses

Graphing functions is an excellent way to visualize their behavior and understand the relationship between functions and their inverses. For a function \( f(x) = x^3 + 1 \), the graph is a cubic curve that is characteristic of cubic functions. This specific one is shifted 1 unit up due to the +1.

The inverse function, \( f^{-1}(x) = \sqrt[3]{x-1} \), is essentially the reflection of the original function over the line \( y = x \). This means that, visually, where the original function goes up, the inverse goes across, and vice versa. The reflection is due to the swapping of x and y coordinates in their definitions.

• When you graph both \( f(x) \) and \( f^{-1}(x) \) in the same coordinate system, they should cross at the line \( y = x \).
• This graphical approach can help you see the symmetry inherent in functions and their inverses, providing insights into how changes to the original function influence its reflection.

Exploring Cubic Functions

Cubic functions are a fascinating area of mathematics, described by equations of the form \( f(x) = ax^3 + bx^2 + cx + d \). The function from the exercise \( f(x) = x^3 + 1 \) is a standard cubic function with a few interesting properties.

With **cubic functions**:

• They can have one or three real roots, depending on the discriminant, but here, thanks to the +1, this particular function has a continuous curve without any sharp turns.
• The graph of a basic cubic function traditionally passes through the origin, but adding +1 shifts the graph upwards, causing it to pass through (0, 1).
• In cubic functions, there are no restrictions on the y-values, meaning their range is \((-finity, \infty)\).

Understanding the behavior of cubic functions allows for better predictions and insights into complex equations in algebra, calculus, and beyond. They're versatile and form the backbone of many mathematical concepts.