Question

\(f(x)=\sqrt[3]{x}, g(x)=-\sqrt[3]{x}-1\)

Short answer

Function \(f(x)\) is the cube root of any input \(x\). Function \(g(x)\) is the same function but with each output negated (hence a reflection) and then subtracted by 1 (a downward movement of the function)

Step-by-step solution

Analyze the function \(f(x)\)

The function \(f(x) = \sqrt[3]{x}\) is a basic cube root function which denotes that each input \(x\) will be evaluated by finding its cube root.

Analyze the function \(g(x)\)

The function \(g(x) = -\sqrt[3]{x} - 1\) involves two transformations on the basic cube root function: reflection and translation. The negative in front of the cube root indicates that the function is reflected about the x-axis, while the -1 at the end denotes downward translation of the function by 1 unit.

Plot The Functions

While this is not a requirement of this exercice, plotting the functions would provide a visual understanding of the transformations that occur from function \(f(x)\) to function \(g(x)\). One will appear as a diagonal straight line from lower left to top right (each point on it being \(x, x^{1/3}\)). The other is exactly the same, but flipped downwards and moved one step down. This presents an easier understanding of how each of these functions would behave.

Key concepts

Function Transformations

A function transformation involves modifying a basic function to achieve a desired form or positioning. In the case of the cube root function, transformations allow us to change its shape and position on a graph. These transformations can include reflections, translations (shifting), stretching, and compressing.

For example, consider the cube root function defined as \(f(x) = \sqrt[3]{x}\), which is a core function with a distinct "s" shaped curve crossing the origin.

• **Basic Transformations**: Reflections and translations change how the graph is oriented and positioned.
• **Complex Transformations**: Stretching (scaling) and compressing alter the steepness or width of the graph.

Understanding these transformations prepares you for more advanced manipulation in algebra and calculus.

Reflection about the x-axis

Reflection of a function over the x-axis inverts the graph vertically. When a function is reflected in this manner, each point of the original function \((x, y)\) becomes \((x, -y)\). This keeps the x-values constant, but reverses the direction of the y-values.

For the function in our original exercise, \(g(x) = -\sqrt[3]{x}\), the negative sign in front of the cube root indicates this reflection. So, \(f(x) = \sqrt[3]{x}\) becomes \(g(x) = -\sqrt[3]{x}\), flipping the graph across the x-axis.

• This changes all positive y-values in \(f(x)\) to negative values in \(g(x)\).
• The graph now appears to be upside down when compared to the original.

Reflecting functions can dramatically change the graph's orientation and is a useful tool in solving equations.

Downward Translation

Translation of a function involves shifting the entire graph without altering its shape. It can be performed vertically or horizontally. A downward translation moves the graph lower along the y-axis by a set number of units.

For the function \(g(x) = -\sqrt[3]{x} - 1\), the term "-1" indicates that the entire graph is shifted one unit downward. This means every point on \(f(x)\) is moved directly down without any horizontal shift.

• Every point \((x, y)\) now aligns at \((x, y - 1)\).
• This shifts the transformation from the reflection further down the y-axis.

Combined with reflection, this downward move provides a clear depiction of how transformations alter a function's position.

Plotting Functions

Plotting functions provides visual clarity on how transformations affect a function graphically. Starting with a basic function like \(f(x) = \sqrt[3]{x}\), each transformation we apply changes its appearance on the Cartesian plane.

To plot the original and transformed functions:

• Plot \(f(x) = \sqrt[3]{x}\) where the curve passes through the origin and demonstrates the increasing cube root values.
• Then add the transformation actions. Reflect \(-\sqrt[3]{x}\) becomes the inverted graph.
• Finally, apply the translation by moving the curve of \(g(x) = -\sqrt[3]{x} - 1\) one unit down.

Graphing these transformations allows you to directly compare the effects of reflection and translation, enhancing understanding through a visual approach. Aim to use graphing tools or software for precise execution and consistent scaling on paper.