Problem 42 - Algebra and Trigonometry

Question

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ h(x)=-e^{x} $$

Step-by-Step Solution

Answer

The graph of \(h(x)=-e^x\) is obtained by reflecting the graph of \(f(x)=e^x\) about the x-axis. The domain of \(h(x)=-e^x\) is all real numbers and the range is all values less than or equal to 0, denoted as (-∞, 0]. The equation of the asymptote is \(y=0\).

Step 1: Understanding the transformation

The function \(h(x)=-e^x\) is just the function \(f(x)=e^x\) multiplied by -1. This means that every \(y\)-value of \(f(x)=e^x\) is simply negated in the graph of \(h(x)\). This is the reflection of the graph of \(f(x)=e^x\) about the x-axis.

Step 2: Sketching the graph

The graph of \(h(x)=-e^x\) can be obtained by reflecting the graph of \(f(x)=e^x\) about the x-axis. The graph of \(f(x)=e^x\) increases as x gets larger and tends to 0 as x gets smaller, but the negative sign for \(h(x)=-e^x\) means it will decrease as x gets larger and tends to 0 as x gets smaller.

Step 3: Equations of the asymptotes

The x-axis is the horizontal asymptote for the function \(h(x)=-e^x\), acting as a boundary line that the function is tending towards but never quite reaching. Therefore, the equation of the horizontal asymptote is \(y=0\). Since the graph extends to both directions of the x-axis without bound, there's no vertical asymptote.

Step 4: Determining the domain and range

The domain of a function is the set of all possible x-values. In the case of \(h(x)=-e^x\), it can accept any real number. Thus, the domain is \(R\) (all real numbers). The range of a function is the set of all possible y-values. For \(h(x)=-e^x\), the range includes all values less than or equal to 0. Thus, the range is (-∞, 0].

Key Concepts

Exponential Functions

Exponential functions are a special type of mathematical functions where the variable appears as the exponent. In its basic form, it is written as \(f(x) = a^x\), where \(a\) is a constant and \(x\) is the variable. A distinctive feature is that the graph of an exponential function either continuously rises or falls. It never levels out, except along its asymptotes.
\(f(x) = e^x\) is a common exponential function, where \(e\) is approximately 2.718, the base of the natural logarithm. This function is crucial in various scientific fields because of its natural occurrence in growth processes.

• Exponential Growth: The function \(e^x\) showcases growth that becomes increasingly steep.
• Exponential Decay: When the exponent \(x\) is negative, the function represents an inverse or decay. This is seen in \(h(x) = -e^x\), indicating a reflection over the x-axis.

Understanding these functions' basic properties is essential to grasp their behavior and different transformations.

Domain and Range

The domain and range of exponential functions give insights into their behavior. The domain of a function includes all possible values of \(x\) for which the function is defined. For exponential functions such as \(h(x) = -e^x\), the domain is all real numbers, symbolized as \( \mathbb{R} \). This means \(x\) can be any real number.
The range of a function refers to all possible values of \(y\) after computing the function. For \(h(x) = -e^x\), the range comprises negative values, specifically \((- \infty, 0]\). The graph of this function will never cross above the x-axis, emphasizing the importance of understanding the range when interpreting graphs.

Graphing Utilities

Graphing utilities are tools, both manual and digital, that aid in visualizing functions accurately and efficiently. These tools include graph paper, calculators, and computer software. For functions like \(h(x) = -e^x\), a graphing utility can confirm a neat, hand-drawn graph by offering a professional version. Here's why they're helpful:

• Accuracy: Automated utilities provide precise plots based on input functions, helping catch any human error.
• Efficiency: It can quickly graph complex transformations, aiding understanding.
• Versatility: Users can experiment with changes to parameters or transformations to see immediate effects on the graph.

Getting comfortable with these utilities complements manual graphing skills, ensuring understanding and accuracy.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. They provide valuable information about the behavior of functions as they extend towards infinity.
For the function \(h(x) = -e^x\), an important feature is its horizontal asymptote at \(y = 0\). This asymptote tells us that as \(x\) moves to positive or negative extremes, the function approaches but never surpasses the horizontal line on the graph.

• Horizontal Asymptote: It indicates the constant value the function nears without actually reaching. For \(-e^x\), this is the x-axis, or \(y = 0\).

Recognizing asymptotes is key in understanding the limits and behavior of exponential functions in real-world contexts. They guide us in projecting the trend of a graph and ensuring interpretations are accurate.