Problem 21 - A Graphical Approach to College Algebra

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 function.

Step 1: Identify the Function Type

The function given is exponential of the form \(f(x) = -1.5^x\). It contains a base of 1.5 and it is multiplied by -1, which will affect the direction of the graph.

Step 2: Determine the Domain

The domain of an exponential function is all real numbers. Therefore, the domain of \(f(x) = -1.5^x\) is \( (-\infty, \infty) \).

Step 3: Determine the Range

For \(f(x) = -1.5^x\), the range is determined by the negative sign in front of the function. Normally, \(1.5^x\) has a range of \( (0, \infty) \), but applying \(-1\) makes the output negative, so the range is \(( -\infty, 0)\).

Step 4: Find the Asymptote

The horizontal asymptote for \(f(x) = -1.5^x\) is \(y = 0\) because as \(x\) approaches infinity or negative infinity, the function approaches zero but never actually reaches it.

Step 5: Determine Increasing or Decreasing

The function \(f(x) = -1.5^x\) is decreasing because the base is greater than 1 and the function is multiplied by -1, causing it to decrease across its domain.

Step 6: Sketch the Graph

Plot a few key points, such as \(f(0) = -1\), \(f(1) = -1.5\), and \(f(-1) = -\frac{1}{1.5}\). The graph starts close to zero on the right and decreases without bound going left. Use a calculator to confirm the shape.

Key Concepts

Domain and Range

When discussing exponential functions like \( f(x) = -1.5^x \), understanding the domain and range is crucial. The **domain** of any exponential function, including this one, comprises all real numbers. This means you can plug any real number into the function, and it will yield an output. This results from the independent variable \(x\) being able to take on any value on the number line, from negative infinity to infinity.

On the other hand, the **range** for this specific function is different because of the negative sign multiplying \(1.5^x\). Normally, without the negative sign, \(1.5^x\) would only produce positive outputs (ranging from zero to positive infinity). However, the negative sign flips this range. Thus, the range of \(-1.5^x\) is \((-\infty, 0)\), meaning all the values are negative, and the outputs never reach zero.

Asymptotes

An important feature of exponential functions is their asymptotes. An **asymptote** is a value that the function approaches but never actually reaches. For the function \( f(x) = -1.5^x \), the asymptote is horizontal and is located at \(y = 0\).

As the \(x\) values increase or decrease without bound, the function \(-1.5^x\) tends towards zero. Despite coming very close, it will not touch or cross the line \(y = 0\). This behavior creates a horizontal asymptote at \(y = 0\).

• As \(x\) approaches positive infinity, \(-1.5^x\) gets closer to zero from the negative side (since it's always negative due to the leading \(-1\)).
• As \(x\) approaches negative infinity, the function value also heads towards zero.

Increasing and Decreasing Functions

In the context of exponential functions, figuring out whether a function is increasing or decreasing is straightforward, but essential. For \( f(x) = -1.5^x \), the function is **decreasing** over its entire domain.

This results from two factors:

• The base, \(1.5\), is greater than one, normally suggesting an increasing pattern.
• However, it is multiplied by \(-1\), flipping the direction of the graph.

These properties together mean that as \(x\) increases, \(f(x) = -1.5^x\) becomes more negative, leading to a decreasing function across all real numbers.

Visualize this behavior by plotting some key points. For example, for \(f(0) = -1\), \(f(1) = -1.5\), and \(f(-1) = -\frac{1}{1.5}\). The graph will move from close to zero on the right side to steeper negatives as you go left on the \(x\)-axis.