Problem 42 - 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\(\pm \infty .\) $$g(x)=-4 e^{2 x}$$

Step-by-Step Solution

Answer

The y-intercept of the function is -4. The domain is all real numbers (\(-\infty < x < \infty\)) and the range is \(-\infty < y \leq 0\). The horizontal asymptote is the line \(y = 0\). As \(x\) approaches positive infinity, the function falls to negative infinity. As \(x\) approaches negative infinity, the function rises towards the horizontal asymptote at \(y = 0\).

Step 1: Identify the function

First, identify the function given. Here, the function \(g(x) = -4e^{2x}\) is an exponential function where \(e\) is the base of natural logarithms, approximately equal to 2.71828, \(x\) is the exponent, and -4 is a constant factor.

Step 2: Calculate the y-intercept

The y-intercept is the value of \(g(x)\) when \(x = 0\). So, substitute \(x = 0\) into the function to find the y-intercept. The result is \(g(0) = -4e^{2*0} = -4e^0 = -4*1 = -4\). So, the y intercept is at (0, -4).

Step 3: Determine the domain and range

For all exponential functions, the domain is all real numbers, so for this function, \(x\) can be any real number (\(-\infty < x < \infty\)). Since the output of the exponential function \(e^{2x}\) is always greater than 0 and it is multiplied by -4, the range of the function is less than or equal to 0. Therefore, the range is \(-\infty < y \leq 0\).

Step 4: Find the horizontal asymptote

An horizontal asymptote is a line that the function gets closer and closer to as \(x\) moves towards positive or negative infinity. As \(x\) tends to infinity, the value of \(e^{2x}\) tends to infinity and -4 times infinity is negative infinity. But as \(x\) tends to negative infinity, the value of \(e^{2x}\) tends to 0 (because any number raised to a very large negative number tends towards 0) and -4 times 0 equals 0. Therefore, the line \(y = 0\) is the horizontal asymptote.

Step 5: Determine the behavior of the function as \(x\) approaches\(\pm \infty .\)

As \(x\) approaches positive infinity, the function \(g(x) = -4e^{2x}\) falls to negative infinity. Conversely, as \(x\) approaches negative infinity, the function rises towards the horizontal asymptote at \(y = 0\).

Key Concepts

Domain and Range

In mathematics, when we talk about the **domain** of a function, we refer to all possible input values (in this case, all the values of \( x \)) for which the function is defined. For exponential functions like \( g(x) = -4e^{2x} \), the domain is particularly simple. It includes all real numbers, so you can input any number for \( x \), and the function will produce an output. This is expressed as:

• Domain: \(-\infty < x < \infty \)

The **range** of a function, on the other hand, is all possible output values (the values of \( y \) you can get). Since this is an exponential function, its behavior is influenced by the exponential part \( e^{2x} \), which is always positive for any real number \( x \). However, in \( g(x) = -4e^{2x} \), the product \(-4 \times e^{2x}\) makes the output negative for any \( e^{2x} \), giving us:

• Range: \(-\infty < y \leq 0\)

This tells us that the graph of the function will only exist below or on the horizontal asymptote \( y = 0 \).

Horizontal Asymptote

A **horizontal asymptote** represents a straight line that the graph of a function approaches as \( x \) moves towards positive or negative infinity. In the context of \( g(x) = -4e^{2x} \), the horizontal asymptote occurs as a result of the inherent properties of its components. As \( x \) approaches negative infinity, the term \( e^{2x} \) converges towards zero since exponentiating a number with a very large negative power leads to a value close to zero. Consequently, the product, \(-4 \times e^{2x}\), also nears zero. This results in the line \( y = 0 \) serving as the horizontal asymptote for this function.Such an understanding provides valuable insights into how graphs appear visually. Here, it's crucial to visualize that the function "hugs" this horizontal line from below, never crossing it, but getting infinitely close as you observe the behavior of the function far to the left. This continual approach without overlap is a defining feature of exponential decay in functions like this one.

End Behavior

**End Behavior** describes how a function behaves as \( x \) approaches extreme values, which, in mathematical terms, means \( x \to \infty \) or \( x \to -\infty \). In our function \( g(x) = -4e^{2x} \), end behavior provides insight into the long-term trends of the graph.

• As \( x \to \infty \): The exponential term, \( e^{2x} \), grows rapidly, resulting in \(-4e^{2x}\) diving down faster towards negative infinity. This means the graph plunges deeply as you move far to the right, indicating steep decline.
• As \( x \to -\infty \): Oppositely, \( e^{2x} \) diminishes towards zero at an accelerated rate, and \( -4 \times 0 \) effectively becomes zero. Hence the function appears to "flatten out" and gently rise to meet the horizontal asymptote \( y = 0 \) from below, without crossing over.

Grasping end behavior helps in drawing graphs that reflect true exponential characteristics in real-world contexts, often seen in depictive scenarios like cooling temperatures, decaying radioactive particles, or depleting resources where something approaches a limit but never quite reaches it.