Part 1
Describe the graphs of the functions f(x) = 2^x – 1 and g(x) = –2^x – 1
Part 2
Compare and contrast the domain and range of f(x) and g(x).
Part 1:
To describe the graphs of the functions f(x) = 2^x – 1 and g(x) = –2^x – 1, we can analyze their behavior and important characteristics.
The graph of f(x) = 2^x – 1 represents an exponential function. When the exponent x increases, the function value increases rapidly. For positive values of x, the output of the function approaches infinity. For negative values of x, the output approaches zero. The graph of f(x) will always be above the x-axis because the minimum value it can reach is -1.
On the other hand, the graph of g(x) = –2^x – 1 is the same shape as the graph of f(x), but it is reflected vertically (upside down) due to the negative sign in front of the 2^x term. This reflection means that g(x) will always be below the x-axis and never intersect it. Like f(x), the maximum value g(x) can reach is -1.
Part 2:
Now we can compare and contrast the domain and range of f(x) and g(x).
Domain: The domain refers to all the possible input values (x) for a function. In this case, both f(x) and g(x) can accept any value for x, because there are no restrictions on the values that can be plugged into the exponential function. Therefore, the domain of both f(x) and g(x) is the set of all real numbers, which can be expressed as (-∞, ∞).
Range: The range represents all the possible output values (y) of a function. For f(x) = 2^x – 1, the range includes all values greater than or equal to -1. As x approaches negative infinity, f(x) approaches -1, but it can go beyond that and increase indefinitely as x becomes positive. So the range of f(x) is (-1, ∞).
On the other hand, for g(x) = –2^x – 1, the range includes all values less than or equal to -1. The exponential term -2^x always produces negative values, and then by subtracting 1, the range extends from negative infinity to -1. Therefore, the range of g(x) is (-∞, -1]. Notice that g(x) can never reach or exceed -1.
In summary, the domain of both f(x) and g(x) is (-∞, ∞), while the range of f(x) is (-1, ∞) and the range of g(x) is (-∞, -1].