Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range. and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=-1.5^{x}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((-\infty, 0)\), Asymptote: \(y=0\), Decreasing on its domain.

Step 1: Identify Function Type

The function given is an exponential function in the form of \(f(x) = -1 imes 1.5^{x}\). This indicates that it is a reflection of the exponential growth function \(1.5^{x}\) across the x-axis due to the negative sign.

Step 2: Determine the Domain

The domain of any exponential function, such as \(-1.5^{x}\), is all real numbers, symbolically \( (-\infty, \infty) \).

Step 3: Determine the Range

For the given function \(f(x) = -1.5^{x}\), the output is always negative since the \(1.5^{x}\) function (which is always positive) is reflected across the x-axis. Hence, the range is \((-\infty, 0)\).

Step 4: Find the Horizontal Asymptote

The horizontal asymptote of an exponential function \(a^{x}\) where \(a>0\) is at \(y = 0\). Since \(-1.5^{x}\) mirrors the function across the x-axis, the asymptote remains \(y = 0\).

Step 5: Identify Increasing/Decreasing Behavior

Since \(f(x)\) is a reflection of an increasing function \(1.5^x\) across the x-axis, the result is a decreasing function on \(( -\infty, \infty)\). All the values of \(f(x)\) decrease as \(x\) increases.

Step 6: Sketch the Graph

On the graph, plot a few points by picking values of \(x\) such as \(x = -2, -1, 0, 1, 2\) to find corresponding \(f(x)\) values. Connect these points showing the reflection across the x-axis. Utilize a calculator to confirm the steepness of the decline and horizontal asymptote at \(y=0\).

Key Concepts

Domain and Range

The domain of a function tells us all the possible input values (x-values) that are allowed. For exponential functions like the one in our example, \(f(x) = -1.5^x\), the domain is quite straightforward. We can put any real number into the place of \(x\), which means the domain is all real numbers, expressed as \(( -\infty, \infty )\).

Now, let's talk about the range, which is all the possible output values (y-values) that the function can produce. With our function \(f(x) = -1.5^x\), the base \(1.5^x\) always gives positive results. However, because of the negative sign in front, all the outputs of \(f(x)\) are negative. Thus, the range is \(( -\infty, 0 )\). This means \(f(x)\) never reaches 0, keeping outputs strictly below it.

Graphing by Hand

When graphing functions by hand, starting with a few key points makes the process easier. For our exponential function \(f(x) = -1.5^x\), it's helpful to choose simple values of \(x\), such as \(-2, -1, 0, 1, 2\).

For \(x = 0\), you will find that \(f(0) = -1.5^0 = -1\) because any number raised to the power of 0 is 1. As you increase \(x\) from 0 to 1, 2, and so on, notice that \(f(x)\) becomes more negative, confirming that the graph decreases as \(x\) increases. Conversely, for negative \(x\)-values like \(-1\) or \(-2\), \(f(x)\) becomes less negative.

Connect your plotted points smoothly, and you will see the graph sloping downwards, highlighting its decreasing nature. Using a calculator can help verify that your sketch accurately represents what’s happening mathematically.

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. For exponential functions, we usually focus on horizontal asymptotes. In the case of \(f(x) = -1.5^x\), this function reflects across the x-axis, just like \(1.5^x\), so its horizontal asymptote is at \(y = 0\).

This means that as \(x\) becomes a large negative number, the value of \(f(x)\) gets closer and closer to 0 but doesn’t actually become 0. Instead, it stays negative, hugging closer to the x-axis. The function never quite touches or crosses the asymptote, illustrating the behavior of exponential decay with a flip in this scenario.

Increasing and Decreasing Functions

Understanding when a function is increasing or decreasing helps in identifying trends and patterns. A function is increasing if its output values become larger as the input values increase. Conversely, it’s decreasing if output values shrink as inputs rise.

For the function \(f(x) = -1.5^x\), it is a reflection of the positive base exponential function \(1.5^x\), which on its own would increase. The negative sign causes \(f(x)\) to decrease across all x-values. In this function, every step to the right on the x-axis makes the y-values go down, making it a decreasing function over its entire domain \(( -\infty, \infty )\).

This decrease is continuous and steady, characterized by an ever-increasing sense of steadiness or strength in the downward slope as you plot your graph to the right. This trait is a key feature of reflected exponential functions.