Problem 95 - Precalculus

Question

For the following exercises, graph the transformation of \(f(x)=2^{x}\) . Give the horizontal asymptote, the domain, and the range. $$h(x)=2^{x}+3$$

Step-by-Step Solution

Answer

Horizontal asymptote: \( y = 3 \), Domain: \( (-\infty, \infty) \), Range: \( (3, \infty) \).

Step 1: Identify the Base Function

The base function given is \( f(x) = 2^x \). This is an exponential function with a horizontal asymptote at \( y = 0 \), domain of all real numbers, and a range of \( (0, \infty) \).

Step 2: Identify the Transformation

The given function is \( h(x) = 2^x + 3 \). This represents a vertical shift of the base function \( f(x) = 2^x \) by 3 units upwards.

Step 3: Graph the Transformation

To graph \( h(x) = 2^x + 3 \), start with the graph of \( f(x) = 2^x \), which passes through the point (0, 1), and shift every point up by 3 units. The new graph will pass through the point (0, 4).

Step 4: Determine the Horizontal Asymptote

For \( h(x) = 2^x + 3 \), the horizontal asymptote of \( f(x) = 2^x \) at \( y = 0 \) is also shifted up by 3 units. Therefore, the horizontal asymptote is \( y = 3 \).

Step 5: Identify the Domain

The domain of \( h(x) \) is all real numbers, \( (-\infty, \infty) \), since exponential functions are defined for all real values of \( x \).

Step 6: Identify the Range

Since \( 2^x \) has a range of \( (0, \infty) \) and it is shifted 3 units up, the range of \( h(x) = 2^x + 3 \) is \( (3, \infty) \).

Key Concepts

Understanding Graph Transformation

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.

Exploring the Horizontal Asymptote

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.

Defining the Domain of the Function

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.

Identifying the Range of the Function

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.