Problem 45 - College Algebra

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

a. The inverse function is \(f^{-1}(x) = \sqrt[3]{x + 1}\). b. Both \(f(x)\) and \(f^{-1}(x)\) can be graphed on the same coordinate system, reflecting each other along the line \(y = x\). c. The domain and range for both \(f(x)\) and \(f^{-1}(x)\) are \((- \infty, + \infty)\)

Step 1: Find the inverse function

Let \(y = f(x)\) for \(f(x) = x^3 - 1\). Swap \(x\) and \(y\) to obtain \(x = y^3 - 1\). Solve this equation for \(y\) to find the inverse function. Adding 1 to both sides we get \(x + 1 = y^3\). Cube rooting both sides, we find that \(y = \sqrt[3]{x + 1}\). Therefore, \(f^{-1}(x) = \sqrt[3]{x + 1}\).

Step 2: Sketch \(f(x)\) and \(f^{-1}(x)\)

The function \(f(x) = x^3 - 1\) is a cube function shifted down by 1 unit. This function increases without bound as \(x\) moves in either positive or negative direction. The inverse function \(f^{-1}(x) = \sqrt[3]{x + 1}\) is a cube root function shifted right by 1 unit. This function also increases without bound as \(x\) moves in either positive or negative direction. Both functions are reflection of each other across the line \(y = x\).

Step 3: Determine the Domain and Range

The domain of a function \(f\) is the set of all possible inputs, or \(x\)-values, for which the function is defined. The range is the set of all possible outputs, or \(y\)-values, the function can produce. For the cubic function \(f(x) = x^3 - 1\), there are no restrictions on \(x\), so the domain is all real numbers, which we write in interval notation as \((- \infty, + \infty)\). For \(f^{-1}(x) = \sqrt[3]{x + 1}\), there are also no restrictions on \(x\), so its domain is also all real numbers or \((- \infty, + \infty)\). The range of both functions is also all real numbers, or \((- \infty, + \infty)\), since both functions can produce any real number as output.

Key Concepts

Graphing Inverse Functions

Understanding the relationship between a function and its inverse is key in algebra. When graphing the inverse of a function, the basic idea is to reflect the graph of the original function over the line where every point is mirrored across the line defined by y = x. This line acts as a mirror, with each point of the function reflecting to form the inverse.

Reflection across the line y = x

When graphing the inverse, each point (a, b) on the original function is swapped to become (b, a) on the inverse function. For the cubic function f(x) = x^3 - 1, its inverse is f^{-1}(x) = \( \root{3}\of{x + 1} \). On a graph, these appear as mirrored images with corresponding points reversed in their x and y values, making graphing them a visual exercise in symmetry.

If one is to graph both f(x) and its inverse, you will notice that they are indeed reflective along this line. An effective way to check if you've graphed the inverse correctly is to ensure that the line y = x passes through the middle of both graphs equally, maintaining this perfect mirror image between them.

Domain and Range of Functions

The concepts of domain and range are crucial when dealing with functions and their inverses. The domain is the set of all possible inputs—a complete listing of all permitted x-values—while the range is the set of possible outputs, all permitted y-values a function can produce.

For Cubic Functions

In our example, the cubic function f(x) = x^3 - 1 can take any real number as an input, hence the domain is all real numbers expressed as \( (- \infty, + \infty) \). The outputs—after the transformation—is also able to produce any y-value, meaning the range is also all real numbers. The same is true for its inverse f^{-1}(x), as the cube root function will accept any real number as input and can output any real number.

This representing of domain and range in interval notation is profound because it succinctly conveys the extensiveness of values that a function can handle without needing to rely on lengthy explanations or unnecessary jargon.

Cubic Functions

Cubic functions, like the one in our exercise f(x) = x^3 - 1, are polynomial functions with the highest power of x being three. These functions have a distinct S-shaped curve, showing a smooth continuous increase or decrease without any sharp turns.

The Nature of Cubic Functions

Their shape makes them uniquely valuable for modeling situations where there is a steady progression of change, such as the growth of a population or the displacement of an object under constant acceleration. Moreover, cubic functions are invertible, which means you can always find an inverse function, as seen with the inverse f^{-1}(x) = \( \root{3}\of{x + 1} \).

It's important to note that cubic functions are not bounded—they can produce outputs that go to infinity in either the positive or negative direction. This signifies that any x-value within the domain provides a legitimate output, reinforcing the lack of restrictions on the domain and range for cubic functions and their inverses.