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Chapter 5: Problem 21

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

Short Answer

Expert verified
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

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

What is the inverse of \(f(x)=-\frac{1}{64} x^3 ?\) (A) \(g(x)=-4 x^3\) (B) \(g(x)=4 \sqrt[3]{x}\) (C) \(g(x)=-4 \sqrt[3]{x}\) (D) \(g(x)=\sqrt[3]{-4 x}\)

The range of the function \(y=a \sqrt{x}\) is \(y \geq 0\).

In Exercises 35鈥�46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=-3 \sqrt{\frac{4 x-7}{3}} $$

Consider the function \(f(x)=-x\). a. Graph \(f(x)=-x\) and explain why it is its own inverse. Also, verify that \(f(x)=-x\) is its own inverse algebraically. b. Graph other linear functions that are their own inverses. Write equations of the lines you graphed. c. Use your results from part (b) to write a general equation describing the family of linear functions that are their own inverses.

\(f(x)=\sqrt{x-3}+3\)
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