Finding the Domain and Range of the Exponential Function f(x) = 2^(-x) + 3

Understanding the Domain and Range of f(x) = 2^(-x) + 3

This article explains how to find the domain and range of the exponential function f(x) = 2^(-x) + 3.

What is Domain? The domain of a function refers to all the possible input values (x-values) for which the function is defined.

What is Range? The range of a function represents all the possible output values (y-values or f(x)-values) that the function can produce.

Analyzing the Function

Let's break down f(x) = 2^(-x) + 3 into two parts:

1. 2^(-x): This is an exponential term where the base (2) is raised to a negative exponent (-x). Exponential functions with positive bases are always defined for all real numbers. Since 2 raised to any power will always be positive, there are no restrictions on the domain for this part of the function.

2. + 3: This is a constant term. Constants do not affect the domain, but they do shift the range.

Determining the Domain

Since the exponential term 2^(-x) is defined for all real numbers, the domain of the entire function f(x) = 2^(-x) + 3 is all real numbers, which can be written as (-∞, ∞).

Determining the Range

1. Range of 2^(-x): As x approaches infinity, 2^(-x) approaches 0 (getting smaller and smaller). As x approaches negative infinity, 2^(-x) approaches positive infinity (getting larger and larger). Therefore, the range of 2^(-x) is (0, ∞), meaning all positive numbers.

2. Range of + 3: The constant term 3 simply shifts the entire range of 2^(-x) upward by 3 units.

3. Combining the Ranges: The overall range of f(x) = 2^(-x) + 3 starts at 3 (because of the constant) and extends to positive infinity. The range is (3, ∞).

In conclusion, the domain of f(x) = 2^(-x) + 3 is (-∞, ∞), and the range is (3, ∞).