Problem 33
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. $$g(x)=2 \cdot 2^{x}$$
Step-by-Step Solution
Verified Answer
The graph of function \(g(x) = 2*2^x\) is a vertical stretch by a factor of 2, of the base function \(f(x) = 2^x\). The asymptote for both functions is the x-axis (y = 0). The domain for both functions is \(-\infty < x < \infty\). Their range is \(0 < y < \infty\).
1Step 1: Graph the Base Function
Firstly, graph the base function, \(f(x) = 2^x\). This is an exponential function and would list to the right as positive \(x\) values increases the value of \(f(x)\) exponentially. The asymptote for this graph is the x-axis (y = 0).
2Step 2: Transform the Base Graph
Next, determine how \(f(x) = 2^x\) is transformed to give \(g(x) = 2*2^x\). Here, \(f(x)\) is just multiplied by 2 to get \(g(x)\). This would stretch the \(f(x)\) function vertically by a factor of 2 to give \(g(x)\). The new graph still has the x-axis as its asymptote.
3Step 3: Determine Domain and Range
From the observed transformations and original function, determine the domain and range. For both \(f(x)\) and \(g(x)\), the domain is \(-\infty < x < \infty\) because there are no restrictions on x-values. The range for \(f(x)\) and \(g(x)\) is \(0 < y < \infty\), because the functions are always greater than 0 (positive for values of \(x\) greater than 0 and approaching 0 but never reaching it for negative values of \(x\)).
4Step 4: Verification Using a Graphing Utility
Finally, confirm the drawn graphs, the asymptotes, and the indicated domain and range using a graphing utility, like a graphing calculator or software.
Key Concepts
Transformations of GraphsDomain and RangeAsymptotesGraphing Utility
Transformations of Graphs
Graph transformations are a powerful tool to understand how changes in equations affect their graphical representation. When dealing with exponential functions like \( f(x) = 2^x \), transformations help us visualize changes such as shifts, stretches, and reflections.
In our exercise, we started with the base exponential function \( f(x) = 2^x \) and transformed it into \( g(x) = 2 \cdot 2^x \). This transformation is called a vertical stretch, where each point on the graph of \( f(x) \) is moved twice as far from the x-axis. This means the graph of \( g(x) \) rises faster than that of \( f(x) \).
Transformations such as these are crucial in graphing, as they show how multipliers affect the growth rate of an exponential function.
In our exercise, we started with the base exponential function \( f(x) = 2^x \) and transformed it into \( g(x) = 2 \cdot 2^x \). This transformation is called a vertical stretch, where each point on the graph of \( f(x) \) is moved twice as far from the x-axis. This means the graph of \( g(x) \) rises faster than that of \( f(x) \).
Transformations such as these are crucial in graphing, as they show how multipliers affect the growth rate of an exponential function.
Domain and Range
Understanding the domain and range of a function provides insight into its behavior and limitations. The domain of a function refers to all possible input values (x-values), while the range represents all possible output values (y-values).
For both the base function \( f(x) = 2^x \) and the transformed function \( g(x) = 2 \cdot 2^x \), the domain is \(-\infty < x < \infty\). This tells us there are no restrictions on what x can be; you can substitute any real number for x.
The range for these functions is \(0 < y < \infty\). Both functions are always positive because exponential functions like these only produce positive output values for real number inputs. As x becomes very negative, both graphs approach but never actually reach zero, thus maintaining their range strictly greater than zero.
For both the base function \( f(x) = 2^x \) and the transformed function \( g(x) = 2 \cdot 2^x \), the domain is \(-\infty < x < \infty\). This tells us there are no restrictions on what x can be; you can substitute any real number for x.
The range for these functions is \(0 < y < \infty\). Both functions are always positive because exponential functions like these only produce positive output values for real number inputs. As x becomes very negative, both graphs approach but never actually reach zero, thus maintaining their range strictly greater than zero.
Asymptotes
Asymptotes are lines that a graph approaches but never actually touches or crosses. They are vital for understanding the behavior of a function at its extremes.
In our exponential functions, \( f(x) = 2^x \) and \( g(x) = 2 \cdot 2^x \), the only asymptote is the x-axis, represented by the line \( y = 0 \).
This horizontal asymptote indicates that as x becomes very negative, the function values get closer and closer to zero, but never quite reach it. Exponential functions like these have horizontal asymptotes because they decrease exponentially as x decreases.
In our exponential functions, \( f(x) = 2^x \) and \( g(x) = 2 \cdot 2^x \), the only asymptote is the x-axis, represented by the line \( y = 0 \).
This horizontal asymptote indicates that as x becomes very negative, the function values get closer and closer to zero, but never quite reach it. Exponential functions like these have horizontal asymptotes because they decrease exponentially as x decreases.
Graphing Utility
A graphing utility is a handy tool for visualizing mathematical functions, providing accuracy that complements hand-drawn graphs. Using a calculator or graphing software, you can plot functions like \( f(x) = 2^x \) and \( g(x) = 2 \cdot 2^x \) to better understand their properties.
Graphing utilities help confirm the appearance and key features of the graph. They display the transformations, the consistent domain and range, and the presence of asymptotes. Furthermore, graphing tools allow you to manipulate the view, zoom in on particular sections, and precisely determine function values at various x-values.
While hand-drawing graphs improve your understanding and intuition, a graphing utility ensures precision, particularly useful when exploring complex functions or verifying solutions.
Graphing utilities help confirm the appearance and key features of the graph. They display the transformations, the consistent domain and range, and the presence of asymptotes. Furthermore, graphing tools allow you to manipulate the view, zoom in on particular sections, and precisely determine function values at various x-values.
While hand-drawing graphs improve your understanding and intuition, a graphing utility ensures precision, particularly useful when exploring complex functions or verifying solutions.
Other exercises in this chapter
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