Problem 36
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. $$ g(x)=e^{x+1} $$
Step-by-Step Solution
Verified Answer
The graph of the function \( g(x) = e^{x+1} \) is a shift one unit to the left of the graph of \( f(x) = e^{x} \). The domain of \( g(x) \) is \(-\infty < x < \infty\), the range is \(0 < y < \infty\), and the asymptote is \( y=0 \). This can be confirmed with a graphing utility.
1Step 1: Identify the Transformation
From \( f(x)=e^{x} \) to \( g(x)=e^{x+1} \), there is a transformation that shifts the graph to the left by one unit because of the '+1' inside the exponent.
2Step 2: Draw the Graph
Shifting the graph of \( f(x)=e^{x} \) one unit to the left, but maintaining the same shape and curvature. Make sure to indicate the asymptote in the drawing.
3Step 3: Identify the Domain, Range and Asymptotes
Since the graph is just shifted but not flipped or stretched, the domain and range remain the same as for \( f(x)=e^{x} \). So the domain is \(-\infty < x < \infty\), the range is \(0 < y < \infty\) and the asymptote is \( y=0 \).
4Step 4: Check with Graphing Utility
At this stage, it is recommended to check your conclusion with a graphing utility, just to confirm that your hand-drawn graph, domain, range, and asymptote are accurately reflected in the digital plot.
Key Concepts
Exponential FunctionsDomain and RangeAsymptotesGraphing Utilities
Exponential Functions
Exponential functions are mathematical functions of the form \( f(x) = b^{x} \), where \( b \) is a positive real number, and \( b eq 1 \). They describe situations where growth or decay increases exponentially. The simplest exponential function used in various calculations is the natural exponential function, \( e^{x} \), where \( e \) is Euler's number, approximately 2.718. This function is notable for its unique properties, including its rate of growth and differentiation characteristics.
Understanding the basics of exponential functions is key to analyzing their transformations. Changes within the exponent can lead to shifts, stretches, or reflections of the graph. For example, adding or subtracting a value inside the exponent, like \( g(x) = e^{x+1} \), results in a horizontal shift. Such nuanced transformations form a significant part of graph manipulation and are vital in applications like population growth, compound interest, and more.
Understanding the basics of exponential functions is key to analyzing their transformations. Changes within the exponent can lead to shifts, stretches, or reflections of the graph. For example, adding or subtracting a value inside the exponent, like \( g(x) = e^{x+1} \), results in a horizontal shift. Such nuanced transformations form a significant part of graph manipulation and are vital in applications like population growth, compound interest, and more.
Domain and Range
The domain and range are important concepts when studying functions. They indicate where the function exists and the output values it can produce. For exponential functions like \( f(x) = e^{x} \), the domain comprises all real numbers because the function is defined for any real input \( x \). This means you can input any number, positive or negative into the function.
The range of \( e^{x} \), however, is more selective; it only covers positive real numbers. This is because \( e^{x} \) never reaches zero or turns negative, as it always grows or decays positively or towards zero. Therefore, its range is \( 0 < y < \infty \).
When a graph transformation occurs, such as a horizontal shift, the domain and range don't change for an exponential function. This is due to the nature of exponential growth, which is not affected by such shifts, maintaining its domain as \(-\infty < x < \infty\) and range as \(0 < y < \infty\).
The range of \( e^{x} \), however, is more selective; it only covers positive real numbers. This is because \( e^{x} \) never reaches zero or turns negative, as it always grows or decays positively or towards zero. Therefore, its range is \( 0 < y < \infty \).
When a graph transformation occurs, such as a horizontal shift, the domain and range don't change for an exponential function. This is due to the nature of exponential growth, which is not affected by such shifts, maintaining its domain as \(-\infty < x < \infty\) and range as \(0 < y < \infty\).
Asymptotes
Asymptotes are crucial in understanding the behavior of graphs, especially for exponential functions. An asymptote is a line that a graph approaches but never touches. For \( f(x) = e^{x} \), there is a horizontal asymptote at \( y=0 \). This means that as \( x \) approaches negative infinity, \( e^{x} \) gets closer and closer to zero, but will never actually reach it.
Understanding the asymptotes helps in sketching and predicting the end behavior of functions. When transformations are applied, such as horizontal shifts, asymptotes often remain unchanged. For example, the function \( g(x) = e^{x+1} \) retains the horizontal asymptote of \( y=0 \) despite the shift of the graph, because the transformation only affects the \( x \)-axis by moving the curve horizontally, not changing its vertical boundaries.
Understanding the asymptotes helps in sketching and predicting the end behavior of functions. When transformations are applied, such as horizontal shifts, asymptotes often remain unchanged. For example, the function \( g(x) = e^{x+1} \) retains the horizontal asymptote of \( y=0 \) despite the shift of the graph, because the transformation only affects the \( x \)-axis by moving the curve horizontally, not changing its vertical boundaries.
Graphing Utilities
Graphing utilities are powerful tools in mathematics for visualizing functions and verifying hand-drawn graphs. These utilities can take various forms, such as graphing calculators, software like Desmos or GeoGebra, and even some advanced scientific calculators.
They simplify the process of graphing complex functions by allowing quick alterations and providing visual confirmations of properties like domain, range, and asymptotes.
For functions like \( g(x) = e^{x+1} \), a graphing utility can precisely confirm transformations such as shifts. It can be especially useful when educational settings demand accuracy, or when hand-drawn results require validation. By plugging the equation into a graphing tool, students can see immediate feedback and ensure their understanding of graph transformations is correct.
They simplify the process of graphing complex functions by allowing quick alterations and providing visual confirmations of properties like domain, range, and asymptotes.
For functions like \( g(x) = e^{x+1} \), a graphing utility can precisely confirm transformations such as shifts. It can be especially useful when educational settings demand accuracy, or when hand-drawn results require validation. By plugging the equation into a graphing tool, students can see immediate feedback and ensure their understanding of graph transformations is correct.
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