In Exercises 19–32, find the (a) domain and (b) range.


𝔂 = cos(x - 3) + 1

Verified step by step guidance

1. Step 1: Understand the function y = cos(x - 3) + 1. This is a transformation of the basic cosine function, where the graph is shifted horizontally by 3 units to the right and vertically by 1 unit upwards.
2. Step 2: Determine the domain of the function. The cosine function, cos(x), is defined for all real numbers. Therefore, the domain of y = cos(x - 3) + 1 is also all real numbers, which can be expressed as (-∞, ∞).
3. Step 3: Analyze the range of the function. The basic cosine function, cos(x), has a range of [-1, 1]. The transformation y = cos(x - 3) + 1 shifts the entire range up by 1 unit.
4. Step 4: Calculate the new range. By shifting the range [-1, 1] up by 1 unit, the new range becomes [0, 2]. This is because the minimum value -1 becomes 0 and the maximum value 1 becomes 2.
5. Step 5: Summarize the findings. The domain of the function y = cos(x - 3) + 1 is (-∞, ∞), and the range is [0, 2].

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function y = cos(x - 3) + 1, the cosine function is defined for all real numbers, so the domain is all real numbers, denoted as (-∞, ∞).

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For y = cos(x - 3) + 1, the cosine function oscillates between -1 and 1, so when shifted up by 1, the range becomes [0, 2].

Transformation of Functions

Transformations involve shifting, stretching, or compressing the graph of a function. In y = cos(x - 3) + 1, the graph of cos(x) is horizontally shifted right by 3 units and vertically shifted up by 1 unit, affecting the range but not the domain.