Graph f(x) = 2^x and g(x) = log2 x in the same rectangular coordinate system. Use the graphs to determine each function's domain and range.

Verified step by step guidance

1. Understand the functions: f(x) = 2^x is an exponential function with base 2, and g(x) = log₂(x) is a logarithmic function with base 2. These two functions are inverses of each other.
2. Graph f(x) = 2^x: Start by creating a table of values for x and f(x). For example, calculate f(x) for x = -2, -1, 0, 1, and 2. Plot these points on the coordinate plane and draw a smooth curve passing through them. Note that the graph will pass through (0, 1) and approach the x-axis as x approaches negative infinity, but it will never touch the x-axis.
3. Graph g(x) = log₂(x): Create a table of values for x and g(x). For example, calculate g(x) for x = 1/4, 1/2, 1, 2, and 4. Plot these points on the coordinate plane and draw a smooth curve passing through them. Note that the graph will pass through (1, 0) and approach the y-axis as x approaches 0 from the right, but it will never touch the y-axis.
4. Determine the domain and range of f(x): The domain of f(x) = 2^x is all real numbers (-∞, ∞), because you can raise 2 to any real number power. The range is (0, ∞), because the output of 2^x is always positive.
5. Determine the domain and range of g(x): The domain of g(x) = log₂(x) is (0, ∞), because you can only take the logarithm of positive numbers. The range is all real numbers (-∞, ∞), because the logarithmic function can produce any real number as an output.

Key Concepts

Exponential Functions

Exponential functions, such as f(x) = 2^x, are mathematical expressions where a constant base is raised to a variable exponent. These functions exhibit rapid growth as x increases and approach zero as x decreases. The domain of an exponential function is all real numbers, while the range is limited to positive values, indicating that the output never reaches zero.

Logarithmic Functions

Logarithmic functions, like g(x) = log2 x, are the inverses of exponential functions. They answer the question of what exponent a base must be raised to in order to produce a given number. The domain of a logarithmic function is restricted to positive real numbers, while the range encompasses all real numbers, reflecting that logarithms can yield any real value.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize their behavior. For f(x) = 2^x, the graph will show exponential growth, while g(x) = log2 x will illustrate a gradual increase that approaches infinity. Understanding how to graph these functions helps in determining their domains and ranges, as well as their intersections and asymptotic behavior.