Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form f(x) = a * b^(x), where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay, depending on the base. In the given function f(x) = 2^(x+3) + 1, the base is 2, indicating that the function will grow exponentially as 'x' increases.
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Domain and Range
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (f(x)). For the function f(x) = 2^(x+3) + 1, the domain is all real numbers, as there are no restrictions on 'x'. The range, however, is limited to values greater than 1, since the minimum value of 2^(x+3) is 0, making the minimum output 1.
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Domain & Range of Transformed Functions
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to visually represent the relationship between the input (x) and output (f(x)). For exponential functions like f(x) = 2^(x+3) + 1, the graph will show a curve that rises steeply to the right and approaches the horizontal line y = 1 as x decreases. Understanding how to graph these functions helps in visualizing their behavior and identifying key features such as intercepts and asymptotes.
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Graphs of Logarithmic Functions