Problem 41 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function and find (a) the \(y\) -intercept; (b) the domain and range; (c) the horizontal asymptote;and (d) the behavior of the function as \(x\) approaches\(\pm \infty .\) $$f(x)=7 e^{x}$$

Step-by-Step Solution

Answer

The graph of function \(f(x)=7e^{x}\) has its y-intercept at y = 7. The domain is all real numbers and the range is from 0 to \(+\infty\)(exclusive of 0). The x-axis (y=0) is the horizontal asymptote. As \(x\) approaches \(+\infty\), the function grows without bounds and as \(x\) approaches \(–\infty\), the function tends to 0.

Step 1: Sketch the function

The function \(f(x) = 7e^{x}\) is an exponential function with base \(e.\) Since base \(e\) is larger than 1, the function grows exponentially as \(x\) increases. The general shape would pass through the point (0, 7) and grows rapidly towards positive infinity. As \(x\) decreases, the function declines rapidly but never touches the x-axis.

Step 2: Compute y-intercept

The y-intercept is the value of the function at x = 0 (i.e., find \(f(0)\)). Substituting \(x = 0\) into the function gives \(f(0) = 7e^{0} = 7\), because \(e^{0} = 1\). So, the y-intercept is at point (0, 7).

Step 3: Finding the Domain and Range

For exponential functions, the domain (values of x) is all real numbers, from \(–\infty\) to \(+\infty\). The range (values of y) is (0, \(+\infty\)) as exponential functions never output negative values or touch the x-axis, but they can output any positive real number.

Step 4: Identify Horizontal Asymptote

A horizontal asymptote is a line that the graph approaches but never crosses. For the function \(f(x) = 7e^{x}\), as \(x\) decreases, the function approaches zero but never reaches it. Thus, the x-axis or y=0 is a horizontal asymptote.

Step 5: Behavior as x approaches \(\pm \infty\)

As \(x\) approaches \(+\infty\), \(f(x) = 7e^{x}\) grows without bound, i.e., it approaches \(+\infty\). As \(x\) approaches \(–\infty\), \(f(x)\) approaches zero (the graph gets closer to the x-axis but never touches it). So, we can say the function tends to 0 as \(x\) goes to \(–\infty\).

Key Concepts

Graph Sketching

When sketching the graph of an exponential function like \( f(x) = 7e^{x} \), it's important to remember that it grows rapidly. The base of the exponential, \( e \), is approximately 2.718, which is greater than 1. This causes the function to increase swiftly as \( x \) becomes larger. The characteristic curve of \( 7e^{x} \) looks like a steeply increasing upward curve.

The graph of \( f(x) = 7e^{x} \) can be sketched by noting its key properties:

• It passes through the \( y \)-intercept at (0, 7) since \( f(0) = 7 \).
• The curve rises steeply to the right as \( x \) approaches positive infinity.
• It asymptotically approaches the x-axis but never touches or crosses it.

By understanding these basic features, you can create an accurate sketch without needing exact calculations for every point.

Domain and Range

To find the domain and range of the exponential function \( f(x) = 7e^{x} \), we consider the values that \( x \) and \( f(x) \) can take. The domain of exponential functions is typically all real numbers, which means \( x \) can be any number from \(-\infty\) to \(+\infty\). This is because you can substitute any real number into \( e^{x} \) to get a valid result.

Now, the range of \( f(x) = 7e^{x} \) is the set of all possible output values. Since \( e^{x} \) is always positive and multiplied by 7, \( f(x) \) cannot be zero or negative. The smallest value it can take is slightly more than zero, but it goes up to positive infinity. Hence, the range is

• Domain: \( (-\infty, +\infty) \)
• Range: \( (0, +\infty) \)

This showcases that the function only outputs positive values, never crossing or reaching zero.

Horizontal Asymptotes

In analyzing horizontal asymptotes of exponential functions, we are interested in the behavior of the function as \( x \) approaches very large positive or negative values. For \( f(x) = 7e^{x} \), as \( x \) approaches negative infinity, the output \( f(x) \) gets smaller and smaller but never actually reaches zero. This line \( y = 0 \) acts as a horizontal asymptote.

An asymptote is a line that the curve gets closer and closer to, but never actually meets or crosses. So, for our function:

• The horizontal asymptote is \( y = 0 \).

This absence of a positive asymptote (like when \( x \) tends to positive infinity) is characteristic of exponential growth functions which rapidly increase, instead of plateauing.

Function Behavior at Infinity

When considering a function's behavior at infinity, we look at what happens as \( x \) moves towards either positive or negative infinity. With \( f(x) = 7e^{x} \), as \( x \) goes to positive infinity, \( 7e^{x} \) grows without bound. This means it increases exponentially, shooting upwards almost vertically.

However, as \( x \) approaches negative infinity, \( f(x) \) approaches zero. This describes a rapidly decreasing function moving closer to the x-axis. Thus:

• As \( x \rightarrow +\infty \), \( f(x) \rightarrow +\infty \).
• As \( x \rightarrow -\infty \), \( f(x) \rightarrow 0 \).

This helps us understand the direction and end behavior of the function. Knowing this behavior aids in grasping not just this function's graph, but also the typical structure of exponential graphs.