Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$5 f(x+1)$$

Step-by-Step Solution

Answer

Domain: \([-2, 1]\), Range: \([0, 15]\).

Step 1: Identify the Transformation on Domain

When we replace \(x\) with \(x+1\), we're actually shifting the domain of \(f(x)\) to the left by 1 unit. The original domain is \([-1, 2]\), so we need to subtract 1 from each part of this interval. Thus, the new domain becomes \([-2, 1]\).

Step 2: Determine the Transformation on Range

The expression \(5f(x+1)\) indicates a vertical stretch of the range of the function \(f(x)\) by a factor of 5. Given that the original range of \(f(x)\) is \([0, 3]\), we multiply each part of this interval by 5. Therefore, the new range becomes \([0 \times 5, 3 \times 5] = [0, 15]\).

Step 3: Summarize the New Domain and Range

Based on the transformations, the domain of \(5f(x+1)\) is \([-2, 1]\), and the range is \([0, 15]\).

Key Concepts

Function Transformation

Transforming functions is a common concept in precalculus that involves manipulating a given function to alter its graph's appearance, such as shifting, stretching, or compressing it. Understanding function transformations can help to visualize how equations behave. Here are some basic ways functions can be transformed:

• Shifting - Moving the function up, down, left, or right on the graph.
• Stretching/Compressing - Making the graph taller, shorter, wider, or narrower.
• Reflecting - Flipping the graph over a line, such as the x-axis or y-axis.

More specifically, when transforming a function like \(f(x)\), we can manipulate the inputs \((x)\), the function itself \((f(x))\), or both, to achieve our desired transformation. Each type of transformation will have a specific impact on the domain and range of the original function.

Vertical Stretch

A vertical stretch occurs when you apply a factor greater than 1 to a function, effectively making it taller. For example, multiplying a function \(f(x)\) by a constant \(a > 1\) will stretch the graph vertically. This transformation affects the range but not the domain.The essential points to remember about vertical stretches are:

• The graph's height increases, making it look taller.
• Only the range changes. Every point on the y-axis is multiplied by the stretch factor.
• It's like zooming in vertically only.

In the exercise, multiplying \(f(x+1)\) by 5 creates a vertical stretch. This changes the range from \([0, 3]\) to \([0, 15]\). Each value from the original range is multiplied by 5, significantly extending the new range.

Horizontal Shift

The horizontal shift is when a function is moved left or right on the graph. This often involves replacing \(x\) in the function \(f(x)\) with \(x\pm c\), where \(c\) dictates the direction and distance of the shift. If \(c\) is positive, the function shifts left; if negative, it shifts right.Key features of a horizontal shift:

• The shift is horizontal. It doesn't affect the height of the function.
• The domain changes, while the range stays the same.
• The graph slides without altering its shape.

In our given problem, replacing \(x\) with \(x+1\) results in a shift 1 unit to the left. Therefore, if the original domain was \([-1, 2]\), after applying the shift, it becomes \([-2, 1]\). This represents how the graph of the function slides but doesn't distort in shape.

Precalculus

Precalculus serves as the foundational course that prepares students for the more advanced concepts of calculus. It predominantly focuses on functions, transformations, and the analysis of different types of functions, such as polynomial, rational, exponential, and trigonometric. We often encounter transformations and manipulations as part of this study, allowing students to understand shifts, stretches, and reflections before tackling calculus. In precalculus, one may focus on:

• Manipulating algebraic expressions and equations.
• Understanding function behavior and transformations.
• Analyzing trigonometric identities and equations.

Mastering precalculus concepts is crucial as they form the building blocks for calculus. By learning about domain and range adjustments through transformations, students will find it easier to comprehend calculus limits, derivatives, and integrals. Concepts like the horizontal shift and vertical stretch provide valuable insights into how functions behave under different mathematical operations.