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Chapter 1: Problem 46

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

Short Answer

Expert verified
\(f(2x)\) is a horizontal compression by a factor of 2.

Step by step solution

01

Identify the Transformation Type

The function given is \( f(2x) \). Here, the transformation involves the input to the function being multiplied by a constant. This is indicative of either a horizontal stretch or compression. Specifically, multiplying the \( x \)-term inside the function by a number greater than 1, such as 2 in this case, results in a horizontal compression.
02

Describe the Horizontal Compression

Since the function is \( f(2x) \), it means that for each original \( x \)-coordinate of \( f(x) \), you need to find \( f \) of twice that \( x \)-value. This compresses the graph towards the y-axis. Essentially, every point on the graph of \( f(x) \) moves closer to the y-axis. If the original point is \( (a, f(a)) \), in \( f(2x) \), it will be \( (a/2, f(a)) \). This effectively halves the distance each input is from the y-axis.

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Key Concepts

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

Horizontal Compression
A horizontal compression occurs when the input of a function is altered by multiplying the variable inside the function by a constant greater than 1. In this case, the function is given as \( f(2x) \). What this means is that each input \( x \) is effectively halved due to the multiplication by 2. This brings every point on the graph closer to the y-axis, squeezing or compressing it horizontally.
  • The original width of the graph is reduced.
  • Points that were further out (greater distance from the y-axis) are now located closer to the y-axis.
The effect can be seen as reducing the range of inputs for the function, compressing the graph's spread.
Original Function
The "original function" refers to the basic form of a function before any transformations are applied to it. In this exercise, the original function is \( f(x) \). This is the function as it exists without any changes to its input or output.
  • It is important to understand how this original form behaves to predict the effects of transformations.
  • The graph of \( f(x) \) provides the baseline from which transformations like horizontal compression are measured.
Before applying any changes, the function \( f(x) \) needs to be understood in its domain, range, and typical behavior.
X-Coordinate
The \( x \)-coordinate is a pivotal element in the transformation of functions, especially in horizontal transformations such as compressions. In the scenario of \( f(2x) \), understanding the \( x \)-coordinate is crucial.
  • Normally in \( f(x) \), a point on the graph might be \((a, f(a))\).
  • Under horizontal compression, this point changes to \((a/2, f(a))\).
Each original \( x \)-coordinate is adjusted, halving the distance it originally held from the y-axis. This focuses on how each horizontal input is affected by the transformation.
Graph Transformation
Graph transformation deals with the changes that occur to a graph based upon modifications within the function's equation. In the case of \( f(2x) \), this means altering the graph of \( f(x) \) in a specific way.The graph undergoes a movement that affects how it is laid out on the coordinate plane:
  • With horizontal compression, the graph appears narrower.
  • Points are moved horizontally towards the y-axis, but vertical positions of points, \( f(a) \), remain unchanged.
Through transforming the graph, the dynamics of your function change, giving you a new perspective on the relationship between input and output values.

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Most popular questions from this chapter

Sketch a graph of each piecewise function. $$ f(x)=\left\\{\begin{array}{cll} x^{2} & \text { if } & x<0 \\ x+2 & \text { if } & x \geq 0 \end{array}\right. $$

Find the domain of each function. $$ f(x)=\frac{x-3}{x^{2}+9 x-22} $$

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

Sketch a graph of each piecewise function. $$ f(x)=\left\\{\begin{array}{ccc} |x| & \text { if } & x<2 \\ 5 & \text { if } & x \geq 2 \end{array}\right. $$

If \(f(x)=\frac{x}{2+x}\) and \(g(x)=\frac{2 x}{1-x},\) find a. \(\quad f(g(x))\) b. \(g(f(x))\) c. What does this tell us about the relationship between \(f(x)\) and \(g(x)\) ?
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