Problem 70 - 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. $$2 f(x-1)$$

Step-by-Step Solution

Answer

The new domain is \([0, 3]\) and the new range is \([0, 6]\).

Step 1: Determine the Transformation of the Domain

Since the function is given as \(2f(x-1)\), we start by considering the transformation \(x - 1\). For the original domain \([-1, 2]\), we solve the inequality \([-1, 2] - 1\). This translates to shifting the domain to the right by 1, resulting in the new domain: \([0, 3]\).

Step 2: Determine the Transformation of the Range

The function is scaled by 2, hence we multiply the original range \([0, 3]\) by 2. Therefore, each value \(y\) in the range is transformed to \(2y\). This results in the new range \([0 \times 2, 3 \times 2]\), which is \([0, 6]\).

Step 3: Verify the Results

After applying each transformation step, verify by checking if the operations affect any constraints of the original function. In this case, shifting the domain and scaling the range affects the function as expected without violating the constraints.

Key Concepts

Domain Transformation

When transforming functions, understanding how the domain is affected is crucial. Imagine the domain as the set of all possible input values for the function. For the function \(2f(x-1)\), the expression \(x-1\) indicates a horizontal shift. Specifically, each value of \(x\) in the domain is shifted to the right by 1 unit.

• Original domain: \([-1, 2]\)
• To account for the shift, solve \(x - 1\) for each boundary, adjusting each value by adding 1.
• New domain: Start \((-1 + 1 = 0)\) to \((2 + 1 = 3)\)
• Transformed domain: \([0, 3]\)

The domain transformation helps in determining where the function is "active." It's like moving the start and end points of a path a little further right.

Range Transformation

Similar to the domain, the range of a function is essential to understand. It includes all possible output values. In the expression \(2f(x-1)\), we focus on the coefficient 2 which scales the range vertically.

• Original range: \([0, 3]\)
• Multiply each output value by 2. This changes the height of the range.
• Calculate: \(0 \times 2 = 0\) and \(3 \times 2 = 6\)
• New range: \([0, 6]\)

By scaling the range, we stretch or shrink the set of possible outputs. In this case, every value is doubled, which extends the range. Range transformation affects how tall or short the function appears on a graph.

Scaling Functions

Scaling functions is like resizing an image. You can make it bigger or smaller, and that's exactly what happens with the function \(y = 2f(x-1)\). Scaling affects both the domain and the range, but the effect depends on whether it is horizontal or vertical.

• Horizontal scaling is indicated by expressions modifying x inside the function (e.g., \(x-1\)). This tends to shift or compress the graph along the x-axis.
• Vertical scaling results from coefficients affecting \(f(x)\), like the 2 in \(2f(x-1)\), stretching or shrinking along the y-axis.

By recognizing these patterns, one can predict how functions will transform graphically. This predictability simplifies thenavigating through algebraic transformations in various functions.