The Exponential Function F(X)=2^X: Domain, Range, Graph, Properties, And Applications.

f(x)=2^x

f'(x)=(2^x)*(ln(2))

Function f(x)=2^x is an exponential function where the base is 2 and the variable is x. Here are some explanations on this function:

1. Domain: The domain of f(x) is all real numbers because we can plug in any real number in place of x and still get a valid result.

2. Range: The range of f(x) is all positive real numbers since 2 raised to any power will always be positive.

3. Graph: The graph of f(x) is an increasing curve that starts from the point (0,1) and goes upward as x increases to infinity. The curve never touches the x-axis, but it approaches it as x approaches negative infinity.

4. Properties: The exponential function f(x)=2^x has some special properties such as:

a. As x approaches infinity, f(x) grows without bound. That is, the function value becomes very large as x becomes very large.

b. As x approaches negative infinity, f(x) approaches zero. This means that the function value becomes very small as x becomes very negative.

c. The function is always positive, so we can say that it is one-to-one and onto.

5. Applications: Exponential functions are often used in real-life applications such as finance, population growth, and radioactive decay. The function f(x)=2^x can be used to model the growth of a population that doubles every year or the decay of a radioactive material that loses half its mass every time interval.

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