Problem 44 - 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. $$ g(x)=\frac{1}{2} e^{x} $$

Step-by-Step Solution

Answer

The function \(g(x)=\frac{1}{2} e^{x}\) is derived from the function \(f(x)=e^{x}\) by a vertical compression by a factor of \(\frac{1}{2}\). The domain of the function is all real numbers while the range is all positive real numbers. It has a horizontal asymptote at \(y=0\).

Step 1: Identify the Transformation

From the given function \(f(x)=e^{x}\), it has been transformed to \(g(x)=\frac{1}{2} e^{x}\). The transformation that occurs here is a vertical compression by a factor of \(\frac{1}{2}\). This is due to multiplying the \(e^{x}\) function by \(\frac{1}{2}\).

Step 2: Determine the Domain, Range and Asymptotes

The domain of the function \(g(x)=\frac{1}{2} e^{x}\) is the set of all real numbers, because \(e^{x}\) exists for all \(x\). The range of the function is all positive real numbers (0,∞) because \(e^{x}\) is always positive and multiplying it by \(\frac{1}{2}\) does not change its sign. The function \(g(x)=\frac{1}{2} e^{x}\) has a horizontal asymptote at \(y=0\) because the function approaches 0 as \(x\) approaches negative infinity (but never actually reaches it).

Step 3: Draw the Graph and Confirm with a Graphing Utility

Drawing this function \(g(x)=\frac{1}{2} e^{x}\) will result in a graph that is similar to \(f(x)=e^{x}\), but it is flatter due to the vertical compression. It will cross the y-axis at \(y=\frac{1}{2}\), rather than at \(y=1\), and it will approach the x-axis as \(x\) gets more negative (forming an asymptote along the x-axis). Additionally, verifying these results using a graphing utility will confirm the drawn graphs.

Key Concepts

Vertical Compression

Vertical compression is a transformation that squeezes a graph towards the x-axis. In the function \( g(x)=\frac{1}{2} e^{x} \), vertical compression occurs because \( e^{x} \) is multiplied by \( \frac{1}{2} \). The original graph of \( f(x)=e^{x} \) has a steep upward curve. By applying a vertical compression, the curve becomes less steep, making it appear flatter.

• This means the graph stretches less vertically.
• It causes each point on the graph to move closer to the x-axis by a factor of \( \frac{1}{2} \).
• Vertical compression does not shift the graph horizontally.

Understanding this concept helps you predict how various transformations will affect a function. With vertical compression, you can make the graph less intense in response to changes.

Domain and Range

The domain of a function is the set of all possible input values (x-values). For \( g(x)=\frac{1}{2} e^{x} \), the domain includes all real numbers. This is because the exponent \( x \) in \( e^{x} \) can be any real number without restrictions.The range of a function is the set of all possible output values (y-values). For \( g(x)=\frac{1}{2} e^{x} \), the range consists of positive real numbers. Since \( e^{x} \) is always positive, and multiplying by \( \frac{1}{2} \) still yields positive values, the range is \((0, \infty)\).

• The graph never touches the x-axis (cannot be zero).
• Values of \( x \) can cause \( g(x) \) to be as large as we want, but never negative.

Knowing the domain and range helps in understanding the behavior of the graph and where it can and cannot go.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or crosses. For exponential functions, horizontal asymptotes usually exist. In \( g(x)=\frac{1}{2} e^{x} \), the horizontal asymptote is at \( y=0 \). This is because as \( x \) approaches negative infinity, \( \frac{1}{2}e^{x} \) gets closer and closer to 0, but never reaches it.

• The closer to \( -\infty \), the closer the function value gets to zero.
• This asymptote is a boundary for how low the function can go, not how high.

Understanding asymptotes helps when predicting end behaviors of functions as they move left or right.

Graphing Functions

Graphing functions, especially exponential ones, involves plotting points and sketching curves based on transformations and properties like domain and asymptotes. For \( g(x)=\frac{1}{2} e^{x} \), follow these steps:

• Start with the basic shape of \( e^{x} \), a smooth curve rising steeply to the right.
• Apply the vertical compression to flatten the curve, so it crosses the y-axis at \( y=\frac{1}{2} \).
• Acknowledge the horizontal asymptote at \( y=0 \) as a guide, showing the graph's tendency as \( x \) decreases.
• Consider using graphing utilities to fine-tune accuracy and confirm your plots.

Graphing requires practice to visualize transformations accurately and to understand interactions between function components.