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 urility to confirm your hand-drawn graphs. $$ g(x)=2 \cdot 2^{x} $$
Step-by-Step Solution
Verified Answer
The function \(g(x)=2\cdot2^{x}\) is a vertical stretch of the function \(f(x)=2^{x}\) by a factor of 2. Both have a domain of (-∞, ∞) and a range of (0, ∞), and both have a horizontal asymptote at y=0. The graphs should have been drawn accordingly and confirmed with a graphing tool if possible.
1Step 1: Graph the Original Function
Begin by graphing the function \(f(x)=2^{x}\). This is an exponential function with a base of 2, which means that it will start at the point (0,1) and rise to the right, while getting closer to but never touching the x-axis (y=0) on the left. Draw the graph appropriately.
2Step 2: Understand the Modification
In the function \(g(x)=2\cdot2^{x}\), the \(2^{x}\) has been multiplied by 2. This doesn't shift the graph left or right, or up or down, but instead vertically stretches it by a factor of 2.
3Step 3: Graph the Modified Function
Draw the graph of \(g(x)\) similar to \(f(x)\), but each y-value will be twice as high. This means that the graph becomes steeper.
4Step 4: Identify the Asymptotes
An asymptote is a line that a graph approaches but never touches. For the function \(f(x)\), its horizontal asymptote is y=0. Since the modification of the function does not involve any vertical shift, the horizontal asymptote for \(g(x)\) will also be y=0.
5Step 5: Find the Domain and Range
The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. Since there are no restrictions on x in the functions \(f(x)\) and \(g(x)\), their domain is (-∞, ∞). As for the range, both functions can take any positive value but never reach 0 or any negative number due to the nature of their exponential growth and the presence of a horizontal asymptote at y=0. Thus, the range for both functions is (0, ∞).
6Step 6: Confirm with a Graphing Tool (optional)
If required or provided, a graphing tool can be used to double-check the hand-drawn graphs. Enter the functions \(f(x)=2^{x}\) and \(g(x)=2\cdot2^{x}\) into the graphing tool and observe the generated graphs. They should align with the understanding of the graphs constructed by hand.
Key Concepts
Asymptotes in Exponential FunctionsDomain and Range of Exponential FunctionsTransformations of Exponential Functions
Asymptotes in Exponential Functions
Understanding asymptotes is crucial in graphing exponential functions. An asymptote represents a line that the graph approaches but never actually reaches. Specifically for exponential functions like \(f(x) = 2^{x}\), the asymptote is a horizontal line that the graph approaches as the value of 'x' decreases indefinitely.
For the function \(f(x) = 2^{x}\), the horizontal asymptote is located at \(y=0\). No matter how far the graph extends to the left (that is, for larger and larger negative values of 'x'), it will never dip below the x-axis. This characteristic defines the behavior and shape of the graph, essentially acting as an invisible boundary.
It's equally important to recognize that transformations, such as scaling or reflections, do not alter the position of a horizontal asymptote if they don't involve vertical shifts. So in the transformed function \(g(x) = 2\cdot2^{x}\), the asymptote remains at \(y=0\). This constancy allows us to predict the behavior of transformed exponential functions with confidence.
For the function \(f(x) = 2^{x}\), the horizontal asymptote is located at \(y=0\). No matter how far the graph extends to the left (that is, for larger and larger negative values of 'x'), it will never dip below the x-axis. This characteristic defines the behavior and shape of the graph, essentially acting as an invisible boundary.
It's equally important to recognize that transformations, such as scaling or reflections, do not alter the position of a horizontal asymptote if they don't involve vertical shifts. So in the transformed function \(g(x) = 2\cdot2^{x}\), the asymptote remains at \(y=0\). This constancy allows us to predict the behavior of transformed exponential functions with confidence.
Domain and Range of Exponential Functions
The domain and range constitute the set of possible inputs (x-values) and outputs (y-values) for a function, respectively. Exponential functions, like \(f(x) = 2^{x}\), typically have a domain that includes all real numbers, meaning you can substitute any real number for 'x', and the function will yield a result. Therefore, the domain for both \(f(x)\) and \(g(x)\) is \((-\infty, \infty)\).
On the other hand, the range is quite different for exponential functions due to their unique properties. Since an exponential function never reaches zero (but rather gets infinitesimally close to it), the range for these functions is restricted to positive numbers. As such, for the functions \(f(x) = 2^{x}\) and \(g(x) = 2\cdot2^{x}\), the range is \((0, \infty)\). Positive exponential growth ensures that the function's value can reach any positive number, but will never reach zero or become negative.
On the other hand, the range is quite different for exponential functions due to their unique properties. Since an exponential function never reaches zero (but rather gets infinitesimally close to it), the range for these functions is restricted to positive numbers. As such, for the functions \(f(x) = 2^{x}\) and \(g(x) = 2\cdot2^{x}\), the range is \((0, \infty)\). Positive exponential growth ensures that the function's value can reach any positive number, but will never reach zero or become negative.
Transformations of Exponential Functions
Transformations alter the graph of a function in specific ways, such as stretching it, compressing it, or shifting it. With the function \(g(x) = 2\cdot2^{x}\), the transformation is a vertical stretch. A vertical stretch occurs when each 'y' value of the original function is multiplied by a constant (in this case, 2), causing the graph to elongate away from the x-axis.
It is vital to grasp that such transformations don't affect the domain of the function or the location of the horizontal asymptote, but they do affect the steepness of the graph. The steeper the graph, the quicker it increases as 'x' gets larger. When interpreting such transformations, it's helpful to think of them as modifying the 'speed' at which the output changes compared to the original function.
To visualize transformations, imagine the base graph \(f(x) = 2^{x}\) as the framework. Any multiplier in front like the 2 in \(g(x) = 2\cdot2^{x}\) stretches the framework vertically. If we were to subtract or add a number inside the exponent, that would move the graph horizontally, but that's not the case in our example.
It is vital to grasp that such transformations don't affect the domain of the function or the location of the horizontal asymptote, but they do affect the steepness of the graph. The steeper the graph, the quicker it increases as 'x' gets larger. When interpreting such transformations, it's helpful to think of them as modifying the 'speed' at which the output changes compared to the original function.
To visualize transformations, imagine the base graph \(f(x) = 2^{x}\) as the framework. Any multiplier in front like the 2 in \(g(x) = 2\cdot2^{x}\) stretches the framework vertically. If we were to subtract or add a number inside the exponent, that would move the graph horizontally, but that's not the case in our example.
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