Problem 75 - 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. $$f(-3 x)$$

Step-by-Step Solution

Answer

Domain: \([-\frac{2}{3}, \frac{1}{3}]\), Range: \([0, 3]\).

Step 1: Understanding the transformation

Given a function transformation of the form \(f(-3x)\), understand that this represents a horizontal transformation because it modifies the input \(x\) by multiplying it by -3.

Step 2: Determine the effect on the domain

The transformation \(-3x\) affects the original domain \([-1, 2]\). Solve for \(x\) in terms of \(y\) using \(y = -3x\). Thus, \(x = -\frac{y}{3}\). Substitute the domain:\([-1, 2]\) becomes \([-\frac{2}{3}, \frac{1}{3}]\) after solving the inequalities \(-3(-\frac{1}{3})\) and \(-3(\frac{2}{3})\).

Step 3: Resolving inequalities for new domain

Transform the inequalities \(-3x = -1\) and \(-3x = 2\) to determine the new limits:1. \(y = -3(-1) \rightarrow x = \frac{1}{3}\)2. \(y = -3(2) \rightarrow x = -\frac{2}{3}\)So the domain is \([-\frac{2}{3}, \frac{1}{3}]\).

Step 4: Effect on the range

Since the transformation does not affect the vertical scaling or shifting, the range remains the same. The original range \([0, 3]\) remains unchanged for \(f(-3x)\).

Step 5: Conclusion on domain and range

The domain of \(f(-3x)\) is \([-\frac{2}{3}, \frac{1}{3}]\) and the range remains \([0, 3]\).

Key Concepts

Domain and Range

When discussing functions, understanding the domain and range is crucial. The domain of a function refers to the set of all possible input values (commonly represented as the x-values) for which the function is defined. For the function \(f(x)\) with an initial domain of \([-1, 2]\), these are the values of \(x\) that we can plug into the function without any issue.
On the other hand, the range is the set of all possible output values (the y-values) that the function can produce. In our example, the initial range is \([0, 3]\), meaning the function can take those x-values given by the domain and return results within 0 and 3.

• Domain: Focuses on input values.
• Range: Focuses on output values.

For the transformed function \(f(-3x)\), the process primarily alters the domain. Due to multiplication by -3, the domain \([-1, 2]\) transforms to \([-\frac{2}{3}, \frac{1}{3}]\), while the range remains unaffected due to no change in the vertical direction. That's why understanding these transformations is fundamental to adjusting both attributes of the function accordingly.

Horizontal Scaling

Horizontal scaling in functions occurs when the input variable \(x\) is multiplied by some constant. The function \(f(-3x)\) implies that the input is scaled horizontally. Multiplying by a negative number like -3 not only compresses the graph horizontally but also reflects it across the y-axis.
In this instance, the function's graph will be squeezed by a factor of 3 and flipped horizontally. This compresses the original domain \([-1, 2]\) to \([-\frac{2}{3}, \frac{1}{3}]\).

• Stretch or Compress: Determined by the absolute value of the constant. A number greater than 1 compresses the graph, while a number between 0 and 1 stretches it.
• Reflection: A negative multiplier causes a reflection across the y-axis.

By comprehensively analyzing the effects of horizontal scaling, students can predict how the function’s graph and its domain will change. This fundamental understanding aids in correctly identifying the new domain bounds.

Graphical Interpretation

Visualizing a function's graph is an excellent way to interpret its behavior under transformation. Graphical interpretation allows students to intuitively grasp the impact of changes to the function's equation.
For \(f(-3x)\), imagine how flipping and compressing affect the graph. If you start with the graph of \(f(x)\), each peak and valley on \(f(x)\) experiences a shift. Not only do x-values become negative, indicating a flip, they are also squeezed closer together, demonstrating a horizontal compression.

• Graph Reflection: Peaks and troughs shift from right to left and vice versa.
• Compression Effects: Diminishes the distance between key graph features along the x-axis.

Understanding these graphical interpretations helps visualize transformations without solving equations manually, making it easier to conceptualize functional changes at a glance.