Problem 68 - 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. $$f(x-3)+1$$

Step-by-Step Solution

Answer

Domain: [2, 5]; Range: [1, 4].

Step 1: Understand the transformation

We need to find the domain and range of the function transformation \(f(x-3)+1\). This transformation involves a horizontal shift to the right by 3 units and a vertical shift upward by 1 unit.

Step 2: Determine the new domain

The original domain of \(f(x)\) is \([-1, 2]\). The transformation \(f(x-3)\) shifts the function 3 units to the right. Therefore, the domain is adjusted by adding 3 to each endpoint of the interval: \([(-1)+3, 2+3] = [2, 5]\).

Step 3: Determine the new range

The original range of \(f(x)\) is \([0, 3]\). The transformation \(+1\) shifts the range upward by 1 unit. Therefore, the range is adjusted by adding 1 to each endpoint of the interval: \([0+1, 3+1] = [1, 4]\).

Key Concepts

Horizontal Shift

A horizontal shift changes the position of a function along the x-axis. This transformation slides the entire graph left or right, without altering its shape or vertical position.
A key point to remember is:

• When the function moves to the right, subtract the shift factor from the input variable inside the function.
• When it moves to the left, add the shift factor.

In our example, the function transformation is represented by \(f(x-3)\), indicating a horizontal shift 3 units to the right. Therefore, every x-value in the domain is increased by 3. If the original domain is \([-1, 2]\), the new domain after shifting is \([2, 5]\). This is calculated by adding 3 to each endpoint of the original domain.

Vertical Shift

Vertical shifts move the graph of a function up or down along the y-axis. This adjustment changes the y-values in the function, while the x-values remain untouched.
Here's how it works:

• When moving upwards, add the shift factor to the entire function output.
• When moving downwards, subtract the shift factor.

In the case of \(f(x-3)+1\), the "+1" indicates a vertical shift upwards by 1 unit. So, if the original range of the function is \([0, 3]\), each value in this interval is increased by 1, resulting in the new range \([1, 4]\). Therefore, the function values are now shifted up by 1, maintaining the same interval width.

Domain

The domain of a function is the complete set of x-values for which the function is defined. Understanding how transformations affect the domain is crucial in solving function-related problems.
In transformations involving horizontal shifts, the entire domain moves along the x-axis. For our scenario, the horizontal shift in \(f(x-3)\) shifts the domain from \([-1, 2]\) to \([2, 5]\).
Always verify the domain post-transformation by considering how the graph has shifted horizontally and ensure that the function is defined in the new interval.

Range

The range of a function is the set of possible y-values it can take. When a function undergoes a transformation, particularly vertical shifts, the range changes accordingly.
In our example, shifting the function \(f(x)-3\) upwards by 1 unit affects the range by moving all y-values one unit higher. Thus, the original range \([0, 3]\) transforms into \([1, 4]\).
To find the range after a shift, adjust each endpoint of the original range by the shift value, ensuring the new range reflects the vertical change in the graph. This ensures that all possible outputs of the function are covered in the specified range.