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Magazine Chapter 1: Problem 6
In Problems 1鈥�6, sketch a graph of the given exponential function. $$ f(x)=\frac{1}{2} 3^{-\sqrt{x}} $$
Short Answer
Expert verified
The graph is an exponentially decaying curve starting at (0, 0.5) and approaching the x-axis as x increases.
Step by step solution
01
Understanding the Function
The given function is \( f(x) = \frac{1}{2} 3^{-\sqrt{x}} \). It is an exponential function where the base is \( 3\) and the exponent is \(-\sqrt{x}\). This indicates that the function will decay as \(x\) increases. Additionally, the factor \(\frac{1}{2}\) scales the function downward.
02
Domain Identification
The function involves \(\sqrt{x}\), implying that \(x\) cannot be negative. Thus, the domain of \(f(x)\) is \(x \geq 0\). The function is defined only for non-negative values of \(x\).
03
Calculating Intercepts
To find the intercepts, calculate \(f(0)\). For \(x = 0\), \(\sqrt{0} = 0\) and \(f(0) = \frac{1}{2} \times 3^{0} = \frac{1}{2} \times 1 = \frac{1}{2}\). Thus, the y-intercept is \(\left(0, \frac{1}{2}\right)\). There is no x-intercept as the function never equals zero for any positive \(x\).
04
Behavior Analysis
As \(x\) increases, \(\sqrt{x}\) increases, hence making \(-\sqrt{x}\) more negative. Therefore, \(3^{-\sqrt{x}}\) approaches zero. Consequently, \(f(x) = \frac{1}{2} \times 3^{-\sqrt{x}}\) also approaches zero as \(x\) increases.
05
Plotting Key Points
Choose a few key values such as \(x = 0, 1, 4\). For \(x=0\), \(f(0) = \frac{1}{2}\). For \(x=1\), \(f(1) = \frac{1}{2} \times 3^{-1} = \frac{1}{6}\). For \(x=4\), \(f(4) = \frac{1}{2} \times 3^{-2} = \frac{1}{18}\). Plot these points on the graph: \((0, \frac{1}{2}), (1, \frac{1}{6}), (4, \frac{1}{18})\).
06
Sketch the Graph
Sketch the curve smoothly through the plotted points considering the exponential decay pattern. The curve starts at \((0, \frac{1}{2})\) and approaches the x-axis but never touches it as \(x\) increases, reflecting an asymptotic behavior towards the x-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graph Sketching
Graph sketching is the process of drawing a curve that represents an equation visually. When it comes to exponential functions like \( f(x) = \frac{1}{2} 3^{-\sqrt{x}} \), you need to consider several aspects to accurately depict their behavior on a graph.
- First, determine key points where the function value is easily calculable. For example, we found earlier that \( f(0) = \frac{1}{2} \), which is a crucial point.
- Next, plot multiple points by calculating values such as \( x = 1 \) and \( x = 4 \), giving us values \( \frac{1}{6} \) and \( \frac{1}{18} \) respectively. These points guide you in sketching the curve.
- Finally, smooth the curve through these points, keeping in mind the decay pattern. The exponential decay means the function decreases rapidly as \( x \) approaches higher values.
Asymptotic Behavior
Understanding asymptotic behavior is key to mastering exponential functions. An asymptote is a line that the graph of a function approaches but never actually reaches. For \( f(x) = \frac{1}{2} 3^{-\sqrt{x}} \), as \( x \) increases, \( \sqrt{x} \) increases.
- This causes \( -\sqrt{x} \) to become more negative, thus making \( 3^{-\sqrt{x}} \) to shrink considerably.
- The function \( f(x) \) approaches zero because it is constantly getting multiplied by smaller values of \( 3^{-\sqrt{x}} \), but it stays positive, suggesting the x-axis is a horizontal asymptote.
Domain Identification
Identifying the domain involves determining all the possible values of \( x \) for which the function is defined. In \( f(x) = \frac{1}{2} 3^{-\sqrt{x}} \), note the presence of \( \sqrt{x} \), which restricts \( x \) to non-negative numbers because square roots of negative numbers are not real.
- Thus, the domain of this function is \( x \geq 0 \), meaning it includes zero and extends to positive numbers.
- This restriction helps in plotting the function, as you only consider these values for sketching and finding points.
Intercept Calculation
Finding intercepts is a fundamental part of analyzing any function. Intercepts mark where the function crosses the axes.
- To find the y-intercept of \( f(x) = \frac{1}{2} 3^{-\sqrt{x}} \), set \( x = 0 \). Calculating, you get \( f(0) = \frac{1}{2} \), so the y-intercept is \( (0, \frac{1}{2}) \).
- There is no x-intercept in this function, as \( 3^{-\sqrt{x}} \) never becomes zero for any positive \( x \). Thus, the graph doesn't cross the x-axis.
