Problem 31 - Algebra and Trigonometry

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 urility to confirm your hand-drawn graphs. $$ g(x)=-2^{x} $$

Step-by-Step Solution

Answer

The graph of \(g(x) = -2^{x}\) is a reflection over the x-axis of the graph for \(f(x) = 2^{x}\). The domain for \(g(x)\) is \(-\infty, \infty\) and the range is \(-\infty,0\). The horizontal asymptote is \(y=0\).

Step 1: Graphing the Base Function: \(f(x)=2^{x}\)

Sketch the graph of the base function \(f(x)=2^{x}\). The range of \(f(x)=2^{x}\) is \(y>0\), and as \(x\) decreases, \(y\) approaches 0 but never reaches it, so \(x=0\) is a horizontal asymptote. As \(x\) increases, \(y\) also increases.

Step 2: Applying the Transformation for the Function \(g(x)\)

The transformation from \(f(x) = 2^{x}\) to \(g(x) = -2^{x}\) is a reflection in the x-axis. This means we take the graph of \(f(x)\) and flip it over the x-axis to find the graph of \(g(x)\). Everything that was above the x-axis for \(f(x)\) is now below the x-axis for \(g(x)\)

Step 3: Graphing the Function \(g(x) = -2^{x}\)

If you graph this function, remember the horizontal asymptote remains the same. As \(x\) decreases, \(y\) goes to 0, so \(y=0\) is a horizontal asymptote. For increasing \(x\) , \(y\) decreases but the graph never crosses \(y=0\).

Step 4: Determining Domain and Range of \(g(x)\)

The domain is the set of all possible x-values and, for this function, any real number can be put for \(x\) so the domain is \(-\infty, \infty\). The range is the set of all possible y-values and, considering our graph, the function never crosses \(y=0\), so all values below 0 are the range. Therefore, the range of the function is \(-\infty,0\).

Key Concepts

Graph Transformations

When you have a function like \(f(x) = 2^x\), you're dealing with an exponential curve that always rises as you move from left to right. But what if we want to modify this base graph? That's where graph transformations come in. The function \(g(x) = -2^x\) is a transformation of \(f(x)\). Here, compare the two: the negative sign in front of the exponential term results in a reflection across the x-axis. This means, everything that was originally above the x-axis in \(f(x) = 2^x\), now flips to below it for \(g(x) = -2^x\). - **Reflection**: The negative sign causes a vertical reflection.- **Vertical Stretch or Compression**: None in this case, as we didn't multiply \(2^x\) by any factor other than -1.Always remember, transformation can drastically affect how a graph looks yet, some core features like asymptotes may remain unchanged. Understanding these changes is crucial in graph analysis.

Domain and Range

The domain and range of a function are fundamental concepts that define where the function exists over x and y. For exponential functions like \(f(x) = 2^x\), these are particularly interesting. **Domain**:- For both \(f(x)\) and \(g(x)\), you can plug any real number into \(x\).- This leads to a domain of \(-\infty, \infty\).**Range**:- For \(f(x) = 2^x\), the output \(y\) is always positive, so the range is \(0, \infty\).- For \(g(x) = -2^x\), due to the reflection, all \(y\) values are negative. Hence, the range becomes \(-\infty, 0\).Knowing the domain gives you the full stretch of x-values, and the range tells you how far up or down the graph will go. Together, these details paint a complete picture of the function’s extent.

Horizontal Asymptotes

Horizontal asymptotes are a key feature in understanding the behavior of exponential functions far out on the x-axis. In our base function \(f(x) = 2^x\), you find the y-values approaching 0 as \(x\) becomes a large negative number. - **For \(f(x)\)**: - As \(x\) decreases, the function \(f(x)\) approaches 0, so \(y = 0\) is a horizontal asymptote. - Graphically, the curve gets closer and closer to the x-axis but doesn’t actually touch it.- **For \(g(x) = -2^x\)**: - Even after reflecting \(f(x)\) over the x-axis, \(y = 0\) remains an asymptote.Understanding this property is very useful when sketching graphs manually. It tells you that no matter how the function changes, how far right or left you go, the function will hover above or below a certain line, never touching it.