Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=e^{x}\) and \(g(x)=\ln (x)\)

Step-by-Step Solution

Answer

Graph the functions together; \( f(x) = e^x \) rises steeply, \( g(x) = \ln x \) rises slowly for \( x > 0 \).

Step 1: Understand the Functions

The given functions are \( f(x) = e^x \) and \( g(x) = \ln(x) \). \( f(x) = e^x \) is the exponential function, which continuously increases for all real numbers. \( g(x) = \ln(x) \) is the natural logarithm function, which is only defined for \( x > 0 \) and increases slowly.

Step 2: Determine Key Points for f(x)

Consider key points for the function \( f(x) = e^x \):- At \( x = 0 \), \( f(0) = e^0 = 1 \).- As \( x \to -\infty \), \( f(x) \to 0 \).- As \( x \to \infty \), \( f(x) \to \infty \).This shows that the graph of \( e^x \) passes through \((0,1)\) and gradually approaches the x-axis as \( x \to -\infty \).

Step 3: Determine Key Points for g(x)

Consider key points for the function \( g(x) = \ln(x) \):- At \( x = 1 \), \( g(1) = \ln(1) = 0 \).- As \( x \to 0^+ \), \( g(x) \to -\infty \).- As \( x \to \infty \), \( g(x) \to \infty \).This indicates the graph of \( \ln(x) \) passes through \((1,0)\) and increases without bound for larger values of \( x \).

Step 4: Sketch the Graphs

On the same set of axes, sketch both graphs. - Plot \( f(x) = e^x \) starting with its key points and showing the curve passing through \((0,1)\) while rising steeply.- Plot \( g(x) = \ln(x) \) showing the graph starting from negative infinity as \( x \to 0^+ \), passing through \((1,0)\), and rising slowly.Both functions highlight the steepness of exponential growth compared to the slower rise of the logarithmic function. Only \( g(x) \) is defined for \( x > 0 \).

Step 5: Analyze the Behavior

Summarize the behavior of both functions:- \( f(x) = e^x \) is always positive and increases rapidly as \( x \) increases.- \( g(x) = \ln(x) \) is undefined for \( x \leq 0 \) and increases but at a decreasing rate as \( x \) increases.Their graphs reflect this behavior prominently. \( f(x) \) dominates for large \( x \).

Key Concepts

Exponential Functions

Exponential functions are a foundational concept in mathematics, especially important when graphing. An exponential function is a type of equation where a constant base is raised to a variable exponent. The most common base is the constant Euler's number, denoted as \( e \), which is approximately 2.718.An example of an exponential function is \( f(x) = e^x \). As we plot the graph of \( f(x) \), we observe its unique behavior:

• For any real number \( x \), the function \( e^x \) is always positive.
• At \( x=0 \), the graph intersects the y-axis at the point \((0,1)\).
• As \( x \) approaches negative infinity, \( f(x) \) approaches 0. This means the graph gets closer to the x-axis but never actually touches it.
• As \( x \) increases towards positive infinity, \( f(x) \) too increases rapidly towards infinity, resulting in a steep upward curve.

The exponential growth demonstrated by \( e^x \) is distinctive for its rate of increase, which itself accelerates as \( x \) increases.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is inversely related to the exponential function. It represents the power to which the base \( e \) must be raised to produce a given number \( x \). Unlike the exponential function, natural logarithms are defined only for positive values of \( x \).Key characteristics of the natural logarithm are:

• \( \ln(x) \) is undefined for \( x \leq 0 \). Hence, its domain is strictly the set of positive real numbers.
• At \( x=1 \), \( \ln(1)=0 \), indicating the point where the graph intersects the x-axis.
• As \( x \) approaches 0 from the positive side, \( \ln(x) \) plunges to negative infinity, creating a vertical asymptote at the y-axis.
• As \( x \) increases, \( \ln(x) \) increases too but at a decreasing rate, making the graph rise smoothly and constantly.

The natural logarithm grows slower compared to an exponential function, making it useful in contexts where exponential growth needs to be linearized.

Function Behavior

Understanding function behavior is crucial for predicting how functions interact on a graph. Exponential and logarithmic functions each exhibit distinctive behaviors.For the exponential function \( f(x) = e^x \):

• This function is always positive and displays rapid growth as \( x \) becomes larger.
• The graph of \( e^x \) rapidly diverges upwards, emphasizing its increase even for small changes in \( x \).

When examining the natural logarithm \( g(x) = \ln(x) \):

• It is undefined for \( x \leq 0 \), confining it to the positive portion of the graph.
• Logarithmic growth is much slower, and as \( x \) increases, the rate of increase continues to decrease.

Comparatively, \( g(x) \) always stays below \( f(x) \) as \( x \) increases, highlighting the dominant behavior of exponential functions over logarithmic ones.

Key Points in Graphs

Key points are vital for sketching accurate graphs, especially when comparing functions like \( f(x) = e^x \) and \( g(x) = \ln(x) \) on the same axes. Identifying these points aids in understanding the interplay between different mathematical functions.For \( f(x) = e^x \):

• The point \((0,1)\) is a crucial intersection with the y-axis.
• Asymptotic behavior near the x-axis as \( x \to -\infty \) suggests the function approaches zero but never reaches it.

For \( g(x) = \ln(x) \):

• The point \((1,0)\) is where the graph crosses the x-axis.
• Vertical asymptote along the y-axis at \( x \to 0^+ \) provides insight into how \( g(x) \) behaves near zero.

Together, these key points and behaviors not only help accurately sketch the functions but also demonstrate the dramatic contrast between how each function grows and behaves.