Problem 69 - A Graphical Approach to College Algebra

Question

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$f(2 x)$$

Step-by-Step Solution

Answer

Domain: \([-1/2, 1]\); Range: \([0, 3]\).

Step 1: Understand the transformation

When considering the function transformation of the form \(f(ax)\), where the function is given as \(f(x)\), the role of \(a\) in \(f(ax)\) is to horizontally stretch or compress the function by a factor of \(\frac{1}{a}\). For \(f(2x)\), with \(a = 2\), the function compresses by a factor of \(\frac{1}{2}\).

Step 2: Compute the new domain

The original function \(f(x)\) has a domain of \([-1, 2]\). When the function is horizontally compressed, we adjust the domain by dividing the endpoints by 2: \(\left[-\frac{1}{2}, 1\right]\). This is because each \(x\) value must produce the same output as when \(x\) lies within \([-1, 2]\).

Step 3: Determine the range

The transformation \(f(2x)\) affects the domain (x-values) but does not affect the range (y-values). Hence, the range remains the same as \(f(x)\), which is \([0, 3]\).

Key Concepts

Domain and Range

The domain and range are fundamental concepts in understanding functions. The domain of a function consists of all the possible input values (typically these are the x-values on a graph), while the range includes all the possible output values (y-values) that result from using the function.
For the function \(f(x)\) with domain \([-1, 2]\) and range \([0, 3]\), we are interested in finding the domain and range for \(f(2x)\). In this case, the domain will adjust due to the horizontal compression, but the range will remain unchanged, as this transformation doesn't affect the output values.

Therefore, the transformed domain of \(f(2x)\) becomes \(\left[-\frac{1}{2}, 1\right]\), calculated by adjusting the original domain endpoints for the transformation factor. However, the range continues to be \([0, 3]\), as the y-values remain unaffected under such transformations.

Horizontal Compression

Horizontal compression occurs when a function's graph is squeezed towards the y-axis. This usually happens when the function is transformed as \(f(ax)\), where \(a > 1\).
For instance, when analyzing the function \(f(2x)\), the function is horizontally compressed by a factor of \(\frac{1}{2}\) compared to \(f(x)\).

This means every point on the graph moves closer to the y-axis. If the untransformed input \(x\) results in an output on \(f(x)\), we now solve for inputs half as large (\(\frac{x}{2}\)) to maintain the same output after transformation.
This compression leads to a narrower domain, calculated by dividing each endpoint of the original domain by \(2\). Ultimately, the domain of \(f(2x)\) becomes \(\left[-\frac{1}{2}, 1\right]\).

Graphical Approach to Functions

Taking a graphical approach to understanding function transformations helps in visualizing how modifications affect the graph's appearance.
When dealing with transformations such as \(f(ax)\), envisioning the changes on the graph can provide clarity.

In the case of \(f(2x)\), a horizontal compression is applied. Imagine gripping the graph on both sides and pushing towards the center. Feature points originally plotted will shrink towards the y-axis, maintaining the form but not the width or spread of the graph.
This visual interpretation aligns with the calculated changes in domain, allowing you to see that although the graph becomes narrower (as confirmed by the new domain \(\left[-\frac{1}{2}, 1\right]\)), the range, or vertical reach, remains \([0, 3]\) unaffected by horizontal compressions.