Problem 15 - Elementary and Intermediate Algebra

Question

a. The function \(f(x)=e^{x}\) is graphed in figure (a) below and the TRACE feature is used. What is the \(y\) coordinate of the point on the graph having an \(x\) -coordinate of \(1 ?\) What is the symbol that represents this number? b. Figure (b) below shows a table of values for \(f(x)=e^{x} .\) As \(x\) decreases, what happens to the values of \(f(x)\) listed in the \(\mathrm{Y}_{1}\) column? Will the value of \(f(x)\) ever be 0 or negative? (GRAPH CANNOT COPY)

Step-by-Step Solution

Answer

a. The y-coordinate at x = 1 is \( e \). b. As x decreases, \( f(x) \rightarrow 0^+ \). \( f(x) \) never reaches 0 or negative.

Step 1: Understand the function

The function given is \( f(x) = e^x \), which is an exponential function. The base of the natural exponential function \( e \) is approximately 2.718. This function is increasing, meaning as \( x \) increases, \( f(x) \) also increases.

Step 2: Finding the y-coordinate for x = 1

We need to evaluate the function \( f(x) = e^x \) at \( x = 1 \). This means we substitute \( x = 1 \) into the function: \( f(1) = e^1 = e \). The \( y \)-coordinate at \( x = 1 \) is thus \( e \), which is a mathematical constant approximately equal to 2.718.

Step 3: Symbol representing the number

The symbol that represents the number \( e \) is \( e \) itself, which is known as Euler's number. It is the base of the natural logarithm and a fundamental constant in mathematics.

Step 4: Analyze behavior as x decreases

According to what we know about exponential functions and based on the table in figure (b), as \( x \) decreases, particularly becoming more negative, the values of \( f(x) = e^x \) decrease towards zero but never actually reach zero.

Step 5: Determine if f(x) reaches zero or negative

For the function \( f(x) = e^x \), the range is all positive real numbers because exponential functions of form \( e^x \) never attain a value of zero or become negative, regardless of how negative \( x \) gets. The asymptote of the graph is the x-axis, which \( f(x) \) approaches but never crosses.

Key Concepts

Euler's Number

Euler's number, commonly represented by the symbol \( e \), is approximately 2.718. It is a fundamental constant in mathematics, much like \( \pi \). We often find \( e \) appearing in many areas of mathematics, especially when dealing with exponential and logarithmic functions.
Just like \( \pi \), \( e \) is an irrational number. This means its decimal expansion goes on forever without repeating. Some key features associated with Euler's number include:

• It is the base of the natural logarithm \( \ln(x) \).
• Its applications span across fields such as calculus, finance, and complex systems.

In the context of the given exercise, \( e \) is used because it is the base of our natural exponential function \( f(x) = e^x \). This property makes \( e \) particularly important when dealing with growth processes.

Natural Exponential Function

A natural exponential function is expressed in the form \( f(x) = e^x \), where \( e \) is Euler's number. This function exhibits unique characteristics and plays a crucial role in mathematical modeling.Some characteristics of the natural exponential function:

• It grows continuously and smoothly as \( x \) increases, never pausing or reversing.
• At \( x = 0 \), the output is \( f(0) = e^0 = 1 \).
• As \( x \) becomes negative, the function approaches zero but doesn't reach it.

This powerful function is used to model compound interest, population growth, radioactive decay, and other phenomena that involve continuous change. In the given problem, calculating \( f(1) = e \) showcases the function's growth properties.

Asymptote Behavior

The concept of asymptotes is vital when discussing the behavior of exponential functions like \( f(x) = e^x \). An asymptote is essentially a line that the graph of a function approaches but never actually touches.For the natural exponential function \( f(x) = e^x \), the horizontal asymptote is:

• The x-axis (\( y = 0 \)) is the line that \( f(x) \) approaches but never reaches.
• As \( x \) tends towards negative infinity, \( e^x \) approaches zero, getting infinitely closer but remaining positive.

This behavior reflects the fact that while \( f(x) \) may get arbitrarily close to zero, it will never become zero or negative. Understanding asymptotic behavior helps in analyzing limits and the long-term behavior of functions.