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

𝔂 = 2e⁻ˣ - 3

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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.

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 (-∞, ∞).

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.

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, ∞).

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, ∞).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

In Exercises 19–32, find the (a) domain and (b) range.𝔂 = 2e⁻ˣ - 3