Problem 32
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. $$g(x)=2^{-x}$$
Step-by-Step Solution
Verified Answer
The function \(g(x) = 2^{-x}\) represents a reflection of the function \(f(x) = 2^{x}\) over the Y-axis. Its graph has an asymptote at y=0. The domain is \(x \in (-\infty , \infty)\) and the range is \(y \in (0, \infty)\).
1Step 1: Understand the transformation
Notice that \(g(x) = 2^{-x}\) is just \(f(x) = 2^x\) with a sign change in the exponent. This is equivalent to reflecting \(f(x)\) with respect to the Y-axis.
2Step 2: Graph the original function \(f(x)\)
Graph the function \(f(x) = 2^x\). It should be clear that the graph passes through the point (0,1), increases as \(x\) increases, and approaches but never reaches the x-axis as \(x\) decreases.
3Step 3: Transform the graph to obtain \(g(x)\)
Reflect the graph of \(f(x)\) over the Y-axis to obtain the graph of \(g(x) = 2^{-x}\). The reflection of the point (a,b) would be the point (-a,b). Therefore, the point (0,1) remains unchanged and for any \(x > 0\) where \(f(x)\) was increasing, \(g(x) = 2^{-x}\) will be decreasing.
4Step 4: Identify the asymptote
The original graph \(f(x) = 2^x\) has an asymptote at the x-axis or y=0. For the transformed function \(g(x) = 2^{-x}\), the asymptote will also be the x-axis or y=0.
5Step 5: Identify the domains and ranges
For the function \(g(x) = 2^{-x}\). The domain (values of x) is all real numbers since we can put any real number into the function. The range (values of y) is y>0. If you look at the graph, the y values are just above the x-axis or y=0, but never touch or cross it. This means that y is always greater than 0.
6Step 6: Validate with a graphing tool
Use any graphing tool to plot the function \(g(x) = 2^{-x}\) to ensure your hand-drawn graph and identified domain and range are correct.
Key Concepts
Exponential FunctionsDomain and RangeAsymptotes
Exponential Functions
Exponential functions are a set of mathematical expressions of the form \( f(x) = b^x \), where \( b \) is a positive constant. These functions exhibit growth or decay behaviors, depending on the base value. In the function \( f(x) = 2^x \), the base is greater than one, which causes the result to increase exponentially as \( x \) becomes larger. This typical growth pattern means the graph rises steeply and accelerates upwards as \( x \) increases.
Exponential decay, on the other hand, occurs when the exponent is negated, as seen in \( g(x) = 2^{-x} \). This transformation flips the growth into decay along the Y-axis. As \( x \) increases, \( g(x) \) decreases rapidly towards zero, creating a mirrored image of the exponential growth. Understanding this transformation helps in visualizing how the graph behaves and how fast it approaches the asymptote as it moves left along the x-axis.
Key points to remember about exponential functions include:
Exponential decay, on the other hand, occurs when the exponent is negated, as seen in \( g(x) = 2^{-x} \). This transformation flips the growth into decay along the Y-axis. As \( x \) increases, \( g(x) \) decreases rapidly towards zero, creating a mirrored image of the exponential growth. Understanding this transformation helps in visualizing how the graph behaves and how fast it approaches the asymptote as it moves left along the x-axis.
Key points to remember about exponential functions include:
- They have a constant ratio between consecutive outputs.
- They demonstrate rapid change — faster than linear or quadratic functions.
- Graphically, they have a continuous curve that never actually touches horizontal asymptotes.
Domain and Range
The domain and range are critical elements of a function, representing all allowable input (\(x\)) and output (\(y\)) values. For exponential functions like \( g(x) = 2^{-x} \), determining the domain and range involves examining its behavior on a graph.
**Domain of Exponential Functions**
For both \( f(x) = 2^x \) and \( g(x) = 2^{-x} \), the domain includes all real numbers. This means any real number value for \( x \) will yield a valid \( y \) outcome. In mathematical terms, the domain is expressed as \( (-\infty, +\infty) \).
**Range of Exponential Functions**
The range, however, is more restricted. For \( g(x) = 2^{-x} \), the output will always be positive, hovering just above zero but never actually reaching it. Therefore, the range is expressed with the inequality \( y > 0 \), meaning all positive real numbers are possible outputs.
Understanding these concepts can clarify what values are permissible inputs and outputs through graphical interpretation and analytical methods.
**Domain of Exponential Functions**
For both \( f(x) = 2^x \) and \( g(x) = 2^{-x} \), the domain includes all real numbers. This means any real number value for \( x \) will yield a valid \( y \) outcome. In mathematical terms, the domain is expressed as \( (-\infty, +\infty) \).
**Range of Exponential Functions**
The range, however, is more restricted. For \( g(x) = 2^{-x} \), the output will always be positive, hovering just above zero but never actually reaching it. Therefore, the range is expressed with the inequality \( y > 0 \), meaning all positive real numbers are possible outputs.
Understanding these concepts can clarify what values are permissible inputs and outputs through graphical interpretation and analytical methods.
Asymptotes
Asymptotes are lines that function graphs approach but never actually reach. In exponential functions, especially ones like \( g(x) = 2^{-x} \), asymptotes play a pivotal role in defining the function's behavior at extreme values.
**Horizontal Asymptotes**
For \( g(x) = 2^{-x} \), the horizontal asymptote is at \( y = 0 \). This means as \( x \) approaches infinity, \( g(x) \) gets closer and closer to the line \( y = 0 \) without ever crossing it. It sets an invisible boundary that \( y \) values approach but never equal.
**Significance of Asymptotes**
They help describe the long-term behavior of the function, showing where the curve flattens out. In practical terms, understanding asymptotes is crucial for predicting limits in calculus and for understanding stability in fields such as physics and economics.
Graphically identifying these lines is key to accurately drawing functions and deducing their behavior at the limits of their domains and ranges. Asymptotes provide clarity about the nature of infinity in exponential functions.
**Horizontal Asymptotes**
For \( g(x) = 2^{-x} \), the horizontal asymptote is at \( y = 0 \). This means as \( x \) approaches infinity, \( g(x) \) gets closer and closer to the line \( y = 0 \) without ever crossing it. It sets an invisible boundary that \( y \) values approach but never equal.
**Significance of Asymptotes**
They help describe the long-term behavior of the function, showing where the curve flattens out. In practical terms, understanding asymptotes is crucial for predicting limits in calculus and for understanding stability in fields such as physics and economics.
Graphically identifying these lines is key to accurately drawing functions and deducing their behavior at the limits of their domains and ranges. Asymptotes provide clarity about the nature of infinity in exponential functions.
Other exercises in this chapter
Problem 31
Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{\sqrt{2}}$$
View solution Problem 32
Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approxi
View solution Problem 32
Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{\sqrt{3}}$$
View solution Problem 32
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
View solution