Problem 28 - College Algebra Essentials

Question

Begin by graphing \(f(x)=2^{x}\). Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$g(x)=2^{x}+2$$

Step-by-Step Solution

Answer

The function \(f(x)=2^{x}\) was graphed with a range of (0,∞) and an asymptote at y=0. The graph of the function \(g(x)=2^{x}+2\) is obtained by shifting the graph of \(f(x)\) upwards by 2 units. Its range is (2,∞) and it has an asymptote at \(y=2\). The domain of both functions is all real numbers (-∞,∞).

Step 1: Graph the Base Function

Begin by graphing the base function \(f(x)=2^{x}\). This function is an exponential function which increases as \(x\) increases. When \(x=0\), \(f(x)=1\), and when \(x=1\), \(f(x)=2\). So, plot these points and sketch the graph. The graph has an asymptote at \(y=0\). The domain of this function is all real numbers (-∞,∞) and the range is (0,∞).

Step 2: Apply the Transformation and Graph the Transformed Function

Next, observe that \(g(x)=2^{x}+2\) is obtained from the base function by shifting it upwards by 2 units. This shift will also shift the asymptote from \(y=0\) to \(y=2\). Graph this function by taking the graph of the base function and shifting every point up by 2 units. The domain of this function is also all real numbers (-∞,∞).

Step 3: Determine the Range of the Transformed Function

The range of \(g(x)\) is determined by its vertical transformation. Since the base function was shifted up by 2 units, the range of the transformed function \(g(x)\) will be \(y>2\), or in interval notation, the range is (2,∞).

Step 4: Confirm the Graphs Using a Graphing Utility

To ensure the graph is correct, it can be confirmed using a graphing utility. By entering the equations for the base function and the transformed function, one should see the graphs match with the hand-drawn graphs.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears as an exponent. The general form of an exponential function is \(f(x) = a^x\), where \(a\) is a constant greater than 0, and \(x\) is the exponent. These functions exhibit distinctive characteristics:

• They grow rapidly, which is beneficial in modeling natural phenomena like population growth or radioactive decay.
• They have a unique shape on a graph, starting slowly, increasing more rapidly as \(x\) becomes larger.

Understanding exponential functions is important because they can transform various types of models and solve equations with growth or decay elements. For instance, in our exercise, we start with \(f(x) = 2^x\), where 2 is the base of the exponential function, representing the constant rate of growth.

Graphing Functions

Graphing functions is the visual representation of mathematical functions on a coordinate plane. It helps in understanding the behavior of the function at different values and provides insights into its characteristics.

• Begin with identifying key points that are easy to compute, such as the y-intercept, which occurs when \(x=0\).
• For \(f(x) = 2^x\), plot points like when \(x=0\), \(f(x)=1\), and when \(x=1\), \(f(x)=2\) to visualize the curve.

Transformations, such as shifts, stretches, or reflections, can alter the graph. In our exercise, the function \(g(x) = 2^x + 2\) is graphed by taking \(f(x)\) and shifting each point up by 2 units. Such transformations are key in understanding how changes to the equation affect the graph, allowing us to predict behavior and features of functions effectively.

Domains and Ranges

The domain and range of functions describe the set of possible inputs (x-values) and outputs (y-values), respectively. Understanding these can help in identifying where the function exists and what it can produce.

• The domain of an exponential function like \(f(x) = 2^x\) is all real numbers \((-\infty, \infty)\). It means any real number can be substituted for \(x\).
• The range is the set of y-values that the function can take. For \(f(x) = 2^x\), it is \((0, \infty)\) because exponential functions never produce zero or negative values.

In the transformed function, \(g(x) = 2^x + 2\), the domain remains the same. However, the range shifts due to the transformation. By adding 2 to \(f(x)\), the range of \(g(x)\) becomes \((2, \infty)\), signifying that the outputs are all greater than 2.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. They are critical in analyzing the behavior of a function as the input values become exceedingly large or small. For exponential functions, horizontal asymptotes are typical features to identify.

• In \(f(x) = 2^x\), the horizontal asymptote is \(y=0\), which the graph approaches but never reaches as \(x\) becomes negative.
• When a function is transformed, such as by shifting upwards, the asymptote also moves in the corresponding direction. With \(g(x) = 2^x + 2\), the asymptote shifts to \(y=2\).

Recognizing asymptotes allows for better predictions about the end behavior of a function. Understanding this concept deeply can aid in sketching accurate graphs and analyzing function traits.