Question

Determine whether the function is one-toone on its entire domain and therefore has an inverse function. $$ f(x)=\sqrt[3]{x+1} $$

Short answer

Yes, the function \( f(x)=\sqrt[3]{x+1} \) is one-to-one and hence has an inverse function.

Step-by-step solution

Determine the Domain

For the function \( f(x)=\sqrt[3]{x+1} \), the domain is all real numbers since you can add 1 to any real number and then take the cube root.

Graph the Function

Next, you should graph the function to visually inspect it. The graph of the given function is a cube root function that has been shifted left by 1 unit.

Apply Horizontal Line Test

Use the Horizontal Line Test on the graph by tracing a horizontal line. If it crosses the graph at more than one point, then the function is not one-to-one. However, for this function, any horizontal line will only intersect the graph once.

Conclude about the function

Based on the Horizontal Line Test, we can conclude that the given function \( f(x)=\sqrt[3]{x+1} \) is a one-to-one function over its entire domain and, therefore, it has an inverse function.

Key concepts

Horizontal Line Test

The Horizontal Line Test is a simple yet powerful visual technique used to determine if a function is one-to-one. A function being 'one-to-one' means that for each value in the function's range, there is exactly one corresponding value in its domain. This property is essential for the existence of an inverse function.

To perform the Horizontal Line Test, imagine drawing horizontal lines through the graph of the function at various heights. If any of these horizontal lines intersect the function's graph more than once, the function is not one-to-one. This happens because two or more inputs (x-values) are producing the same output (y-value), which violates the definition of a one-to-one function.

For instance, in the textbook exercise, applying the Horizontal Line Test to the cube root function results in every horizontal line crossing the graph at only one point, confirming it is one-to-one. By understanding and applying this test, students can confidently determine the one-to-one nature of functions and thereby the existence of their inverses.

Cube Root Function

The cube root function is expressed as \( f(x) = \sqrt[3]{x} \) and is the inverse operation of cubing a number. For example, if \( 2^3 = 8 \), then \( \sqrt[3]{8} = 2 \). Unlike the square root function, which only accepts non-negative inputs (since you can't normally take the square root of a negative number without involving complex numbers), the cube root function can handle all real numbers, including negatives. This is because any real number, whether positive or negative, can be cubed, and therefore has a real cube root.

In the given problem, the function \( f(x) = \sqrt[3]{x+1} \) represents a horizontally shifted cube root function, where the shift does not affect its one-to-one nature. The graph of a standard cube root function has neither horizontal nor vertical asymptotes, and it displays a distinctive 'S'-shaped curve. Such functions are continuously increasing or decreasing, ensuring that each output is unique to one input—an essential feature of one-to-one functions.

Inverse Function

An inverse function, in essence, reverses the action of the original function. If we have a function 'f' that takes an input 'x' and produces an output 'y', then the inverse function, denoted as \( f^{-1} \), takes 'y' as input and gives back 'x'. However, not all functions have inverses. For a function to have an inverse, it must be one-to-one, as each input should map to a unique output and vice versa.

In the context of the provided exercise, since the function \( f(x) = \sqrt[3]{x+1} \) passes the Horizontal Line Test, it confirms that it is indeed one-to-one, and therefore, an inverse function exists. To compute the inverse, one must interchange the roles of 'x' (input) and 'y' (output) and then solve for the newer 'x'. The inversion process is a critical concept in understanding how functions operate, especially when exploring the intrinsic link between a function and its inverse. These relationships are fundamental in algebra, calculus, and many applications in the sciences.