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Chapter 1: Problem 42

Describe how each function is a transformation of the original function \(f(x)\). $$ f(-x) $$

Short Answer

Expert verified
\( f(-x) \) is a reflection of \( f(x) \) over the y-axis.

Step by step solution

01

Understand the Original Function

Let's start by considering the original function \( f(x) \). This function is the base, and transformations are applied to it.Notably, if \( f(x) \) represents a graph, then it is plotted with \( x \) on the horizontal axis and \( f(x) \) on the vertical axis.
02

Identify the Transformation

The expression \( f(-x) \) suggests that the input variable \( x \) has been replaced by \( -x \). This form denotes a horizontal transformation of the original graph.
03

Interpret the Transformation

Replacing \( x \) with \( -x \) in \( f(x) \) results in reflecting the function across the y-axis.This is because each positive value of \( x \) in \( f(x) \) becomes a negative value in \( f(-x) \), and vice-versa, flipping the entire graph over the vertical axis.

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Key Concepts

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

Horizontal Reflection
A horizontal reflection occurs when we change the input of a function from \( x \) to \( -x \). This transformation is often denoted as \( f(-x) \). It's a fascinating phenomenon that flips the graph of \( f(x) \) over the y-axis.
To understand why this happens, let's dive into the behavior of the graph:
  • If you have a point \((a, b)\) on the graph of \( f(x) \), then after applying \( f(-x) \), this point transforms to \((-a, b)\).
  • The effect is that every point from the right side of the y-axis moves to the left side, and every point from the left side moves to the right side.
By visualizing the function this way, it becomes easier to grasp how horizontal reflections work and how they affect the symmetry of the graph.
Graphical Transformations
Graphical transformations form an integral part of understanding functions in mathematics. When we talk about transformations, we refer to the modifications that alter the visual appearance of a graph.
These transformations include:
  • Translation: Shifting the graph vertically or horizontally without changing its shape.
  • Stretching/Compressing: Altering the graph's size, making it taller, wider, or squished.
  • Reflections: Flipping the graph over an axis, either horizontally or vertically.
Each of these transformations changes how the graph looks but retains the core shape and relationship between the variable \( x \) and its output \( f(x) \).
Specifically, a horizontal reflection, as discussed, is a type of reflection that shifts the function across the y-axis. Recognizing these transformations equips students with the tools to manipulate and understand functions more deeply.
Y-axis Reflection
Reflections across the y-axis are special types of transformations applied to functions. When a function \( f(x) \) is transformed into \( f(-x) \), this signifies a y-axis reflection.
This reflection has tangible effects on the graph:
  • Every point on the graph that was originally at \((x, y)\) will now be at \((-x, y)\).
  • The resulting graph is a mirror image across the y-axis.
The concept is essential as it maintains the original function's shape but alters its orientation. Recognizing a y-axis reflection is straightforward: just look for the negative sign inside the argument of \( f(-x) \).
This transformation doesn't just change the graph visually; it also modifies the interpretation of the function mathematically, especially in analyzing symmetry and periodicity.

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Most popular questions from this chapter

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\) $$ h(x)=4+\sqrt[3]{x} $$

Describe how each function is a transformation of the original function \(f(x)\). $$ f(2 x) $$

Describe how each function is a transformation of the original function \(f(x)\). $$ f\left(\frac{1}{5} x\right) $$

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\) $$ h(x)=\frac{3}{x-5} $$

For each function below, find \(f^{-1}(x)\). $$ f(x)=2-x $$
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