Problem 26 - College Algebra Essentials

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. $$g(x)=2^{x+2}$$

Step-by-Step Solution

Answer

The function \(g(x)=2^{x+2}\) is a horizontal shift of the base function \(f(x) = 2^x\) 2 units to the left. The domain of \(g(x)\) is all real numbers and the range is \((0,+\infty)\). The equation of the new asymptote is \(x = -2\).

Step 1: Understanding Translations of Graphs

The given function \(g(x) = 2^{x+2}\) is a transformation of the base function \(f(x) = 2^x\), specifically, a horizontal shift. As \(x\) in the exponent has been replaced by \(x+2\), this dictates a horizontal shift of 2 units to the left.

Step 2: Graph the Transformations

Now graph \(f(x) = 2^{x}\) and \(g(x) = 2^{x+2}\), remembering that the transformation is a horizontal shift of 2 units to the left. This moves the vertical asymptote \(x = 0\) of \(f(x)\) to \(x = -2\) for \(g(x)\).

Step 3: Determine the Domain and Range

From the graphs we obtain that the domain, which is the set of all possible x-values, for both \(f(x)\) and \(g(x)\) is \(R\) (all real numbers). This arises from the fact that we can insert any real number \(x\) into the exponent and get a real number \(f(x)\) or \(g(x)\) out of it. The range, however, has changed due to the shift: for \(f(x)\) it was \((0,+\infty)\), whereas now for \(g(x)\) it is \((0,+\infty)\). Here, the shift has not affected the range, as we are dealing with a horizontal shift, which impacts the domain rather than the range.

Step 4: Use a Graphing Utility to Confirm the Graphs

Use a graphing calculator or a digital tool to verify the drawn graphs. Normally, the results should coincide.

Key Concepts

Exponential Functions

Exponential functions are a fundamental type of mathematical function that involve exponents applied to variable bases. These functions usually take the form of \(f(x) = a^{x}\), where \(a\) is a constant and \(x\) is the variable. Exponential functions have various interesting properties. They are characterized by:

• Rapid growth or decay depending on the value of the base \(a\).
• A consistent rate of change, meaning the function increases or decreases by a fixed percentage at each step.
• The base \(a\) is greater than 0 and not equal to 1, allowing the function to represent repetitive doubling or halving.

If \(a > 1\), the function exhibits exponential growth, accelerating as \(x\) values increase. Conversely, if \(0 < a < 1\), the function exhibits exponential decay. In the base function \(f(x) = 2^x\) for instance, \(f(x)\) grows rapidly since the base, 2, is greater than 1.
Understanding these properties is crucial when dealing with graph transformations, as they provide the basis from which changes and shifts are observed.

Horizontal Shifts

Horizontal shifts are a type of graph transformation that involves moving a function left or right on the x-axis. This shift is determined by changes to the variable \(x\) inside the function. In the context of exponential functions, a shift is represented as \(g(x) = f(x + d)\), where \(d\) is the magnitude of the shift.
For example, in the transformation from \(f(x) = 2^x\) to \(g(x) = 2^{x+2}\), the graph of \(f(x)\) shifts 2 units to the left. This is because replacing \(x\) with \(x + 2\) effectively counteracts with a leftward move.

• If \(d > 0\), the shift moves left.
• If \(d < 0\), the shift moves right.

Horizontal shifts do not alter the shape of the graph; they only change its position horizontally. It's crucial to understand this concept, especially when dealing with functions having asymptotes or specific domain and range characteristics, since it affects how these features are interpreted after the transformation.

Domain and Range

In the study of functions, understanding the domain and range is paramount. The domain of a function is the complete set of possible input values (\(x\)-values), while the range is the realistic output values (\(y\)-values) that the function can produce.
For the exponential function \(f(x) = 2^x\), the domain is all real numbers \(\mathbb{R}\) because you can substitute any real number for \(x\). Exponential functions are continuous everywhere on the x-axis. This applies equally to transformations like \(g(x) = 2^{x+2}\), where the domain remains \(\mathbb{R}\).
The range of \(f(x) = 2^x\) is \((0, \infty)\), as exponential functions always produce positive values, never reaching or touching zero. Despite the horizontal shift that occurs in \(g(x)\), its range is unaffected because the transformation does not impact the function's output values directly.
Understanding these characteristics allows you to better grasp how transformations alter graphed functions without fundamentally changing core properties like domain and range.

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. They help describe the behavior of functions as the input (or \(x\)) grows exceedingly large or small. Exponential functions typically have a horizontal asymptote, often the x-axis, since they tend to level off instead of decreasing to zero.
For the base function \(f(x) = 2^x\), the horizontal asymptote is \(y = 0\). Because exponential functions can grow very large but not infinitely so, they move closer to but do not cross this line.
When a function undergoes a horizontal shift, like in the case of \(g(x) = 2^{x+2}\), the asymptote remains unchanged at \(y = 0\). It's essential to recognize that while horizontal shifts move the graph left or right, they do not affect the asymptote's position.
This comprehension of asymptotes provides valuable insights into the limits and behavior of the function, as well as clearer expectations when sketching or evaluating various transformations.