Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$y=-\sqrt[3]{x+5}$$

Step-by-Step Solution

Answer

The function is one-to-one.

Step 1: Understand the Function

The given function is \( y = -\sqrt[3]{x+5} \). This is a cube root function that has been reflected over the x-axis. Cube root functions are of the form \( y = \sqrt[3]{x} \) which inherently have specific properties.

Step 2: Check for One-to-One Property

A function is one-to-one if every horizontal line intersects the graph of the function at most once. For \( y = -\sqrt[3]{x+5} \), apply the horizontal line test: the horizontal line \( y = c \) intersects the cube root function in only one point for all \( x \). This is due to the fact a cube root is a one-to-one function. Thus, the transformed function remains one-to-one.

Key Concepts

Cube Root Function

A cube root function is a type of function that involves the cube root of a variable, usually in the form of \( y = \sqrt[3]{x} \). These functions are interesting because they have a special kind of symmetry and characteristics. One of the most notable features of the cube root function is that it can take both negative and positive values of \( x \), giving corresponding real numbers as outputs. This makes the cube root function different from square root functions, which only deal with non-negative outputs.

• The graph of a basic cube root function looks like an elongated S-shape, which extends infinitely in both vertical directions.
• Since the cube root of any number exists in the real number system, these functions do not have domain restrictions.

In the function \( y = -\sqrt[3]{x+5} \), the cube root function is shifted horizontally by 5 units to the left and reflected over the x-axis. This means it retains all cube root properties but is inverted along the x-axis.

Horizontal Line Test

The horizontal line test is a useful tool for determining if a function is one-to-one. If, for a given function, every horizontal line intersects the graph at most once, the function is one-to-one. This means that for all \( y \) values, there is exactly one \( x \) value, ensuring each input maps uniquely to one output.

• One-to-one functions are essential for ensuring that functions have inverses that are also functions.
• For the cube root function \( y = -\sqrt[3]{x+5} \), since cube root functions are naturally one-to-one, even with transformations, they will pass the horizontal line test.

Therefore, the transformed cube root function will also be one-to-one, as its basic properties are unchanged by transformation, only shifted and reflected.

Function Transformation

Function transformation involves altering the basic position or shape of the graph of a function through processes such as translation, reflection, stretching, and compression. These transformations allow us to manipulate the graph of the function to produce various effects without changing the underlying relationship between variables.

• Translation: Moves the graph horizontally or vertically. For \( y = -\sqrt[3]{x+5} \), a horizontal shift of 5 units left is applied.
• Reflection: Flips the graph across the axes. In our function, the graph is reflected over the x-axis, inverting its orientation.
• Stretching/Compression: Alters the steepness or width of the graph; however, this is not applied in our example.

These transformations do not affect the one-to-one nature of the original cube root function, meaning that after transformation, the function still maintains this property.