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Chapter 1: Problem 45

Describe how each function is a transformation of the original function \(f(x)\). $$ f(5 x) $$

Short Answer

Expert verified
The function \( f(5x) \) represents a horizontal compression of \( f(x) \) by a factor of 5.

Step by step solution

01

Understand the Original Function

The original function is given as \( f(x) \). This is our base function, and any changes to it, such as scaling or shifting, need to be described in terms of how they relate to \( f(x) \).
02

Identify the Transformation

Observe the function \( f(5x) \). This indicates that the argument (the x-term) inside the function is multiplied by 5. This transformation is a horizontal compression of the function.
03

Explain the Horizontal Compression

For the function \( f(5x) \), the x-values must be multiplied by \( \frac{1}{5} \) to achieve the same output as \( f(x) \). Therefore, this transformation compresses the function horizontally by a factor of 5. Points on the graph will move closer to 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
In function transformations, horizontal compression involves compressing the graph of a function along the x-axis. This happens when a function is transformed from its original form to a new form where the x-term is multiplied by a constant. In the case of the function transformation \(f(5x)\), we are seeing a horizontal compression.
  • This compression factor is determined by the coefficient inside the function's argument, which is 5 in this example.
  • The x-values must be adjusted by the reciprocal of this factor to maintain the functionality of the original equation.
Accordingly, each x-value in the original function must be multiplied by \(\frac{1}{5}\), pulling the points closer to the y-axis. This modification makes the graph appear as if it has been squeezed from the sides.
Original Function
The original function, denoted as \(f(x)\), serves as the base function from which transformations are derived.
  • This base function could represent any kind of mathematical relationship such as linear, quadratic, or trigonometric functions.
  • Transformations of this original function involve either shifting, reflecting, or scaling.
The transformations describe how the new function, like \(f(5x)\), is related to \(f(x)\). These changes are always described based on the foundation provided by the original function. Understanding \(f(x)\) is crucial, as it acts as an unaltered reference point for observing changes like compressions, stretches, or shifts.
Scaling Functions
Scaling a function is all about multiplying or dividing the coordinates to stretch or compress its graph. There are two types of scaling: horizontal and vertical. When discussing the transformation \(f(5x)\), we are concerned with horizontal scaling.
  • Horizontal scaling affects the x-values, adjusting them by a factor either less than or greater than 1.
  • When a function is compressed horizontally by a factor, like 5 in this case, it simplifies to pulling the graph closer to the center along the x-axis.
This kind of transformation alters how quickly the function progresses, making the graph traverse quicker through its values. It's important because it changes the perception and interpretation of the function's growth or decay.

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