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Graph transformation involves altering the basic appearance of a graph in various ways to create a new graph. When dealing with exponential functions like \(f(x) = 2^x\), it is common to perform vertical shifts, horizontal shifts, stretches, or compressions. In this instance, the transformation is a **vertical shift**.

We start with our base function, \(f(x) = 2^x\), and we shift it upwards by 3 units to become \(h(x) = 2^x + 3\). This means every point on the graph of \(f(x)\) moves up by 3 units.

When graphing, check the new position of some key points, especially where they cross the axes. For example, the point \((0,1)\) on \(f(x)\) moves to \((0,4)\) on \(h(x)\), by adding 3 to the y-coordinate.

A horizontal asymptote is a horizontal line that a graph approaches but never actually touches. For the base function \(f(x) = 2^x\), the horizontal asymptote is at \(y = 0\), indicating that as \(x\) approaches negative infinity, the function value gets closer to zero.

With the transformation to \(h(x) = 2^x + 3\), this asymptote shifts upwards by 3 units. Thus, the horizontal asymptote of \(h(x)\) is at \(y = 3\).

This indicates that no matter how large or small \(x\) becomes, the value of \(h(x)\) will approach 3 but never fall below it. Asymptotes are essential in understanding the end behavior of functions, especially for sketching accurate graphs.

The domain of a function is the set of all possible x-values that will produce a valid y-value. For exponential functions like \(f(x) = 2^x\), there is no restriction on \(x\). It can take any real number because \(2^x\) is defined for all real \(x\) values.

Therefore, whether you consider the base function \(f(x)\) or the transformed function \(h(x) = 2^x + 3\), the domain remains unchanged.

• The domain of \(h(x)\) is \((-\infty, \infty)\).

Understanding the domain helps in determining where a function is applicable or analyzable.

The range of a function refers to the set of possible y-values that the function can take. For the exponential function \(f(x) = 2^x\), the outputs are always positive numbers, starting just above zero and extending upwards without bound. Thus, its range is \((0, \infty)\).

After the transformation to \(h(x) = 2^x + 3\), the range of the function shifts upward by 3 units. Hence, the smallest value it can take is now just above 3 instead of just above 0.

• The range of \(h(x)\) is \((3, \infty)\).

Identifying the range is crucial for understanding the full behavior and constraints of a function.