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


𝔂 = 2e⁻ˣ - 3

Verified step by step guidance

1. Step 1: Understand the function given, which is 𝔂 = 2e⁻ˣ - 3. This is an exponential function where the base is e (Euler's number) and the exponent is -x.
2. Step 2: Determine the domain of the function. The domain of an exponential function is all real numbers because you can substitute any real number for x without restriction. Therefore, the domain is (-∞, ∞).
3. Step 3: Analyze the behavior of the function to find the range. As x approaches positive infinity, e⁻ˣ approaches 0, making 𝔂 approach -3. As x approaches negative infinity, e⁻ˣ becomes very large, making 𝔂 approach positive infinity.
4. Step 4: Conclude the range based on the behavior of the function. Since 𝔂 approaches -3 but never actually reaches it, and can increase without bound, the range is (-3, ∞).
5. Step 5: Summarize the findings: The domain of the function is all real numbers (-∞, ∞), and the range is all real numbers greater than -3, which is (-3, ∞).

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 = 2e⁻ˣ - 3, the exponential function e⁻ˣ is defined for all real numbers, meaning the domain is all real numbers, or (-∞, ∞). Understanding the domain is crucial for determining where the function can be evaluated.

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 the function y = 2e⁻ˣ - 3, as x approaches infinity, e⁻ˣ approaches 0, making y approach -3. As x approaches negative infinity, y approaches positive infinity. Thus, the range is (-3, ∞). Knowing the range helps in understanding the behavior of the function.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * bˣ, where a is a constant, b is a positive real number, and x is the exponent. In the given function y = 2e⁻ˣ - 3, the base e (approximately 2.718) is a natural constant, and the function exhibits rapid growth or decay. Understanding the properties of exponential functions is essential for analyzing their behavior and transformations.