Problem 81 - Precalculus

Question

Graph the function \(f(x)=\left\\{\begin{array}{ll}-a^{x} & x < 0 \\ -a^{-x} & x \geq 0\end{array} \quad \text { where } 0 < a < 1\right.\)

Step-by-Step Solution

Answer

The graph consists of two parts with a discontinuity at \(x = 0\). For \(x < 0\), the curve approaches 0 from below. For \(x \geq 0\), the curve starts at -1 and descends sharply.

Step 1: Understand the Function's Piecewise Definition

The function is defined differently based on whether the input value \(x\) is less than 0 or greater than or equal to 0. For \(x < 0\), the function is \(-a^x\). For \(x \geq 0\), it is \(-a^{-x}\).

Step 2: Analyze the Function for \(x < 0\)

For \(x < 0\), the function \(-a^x\) is an exponential function with a negative base, flipped across the x-axis. Since \(0 < a < 1\), as \(x\) becomes more negative, \(a^x\) approaches zero, causing \(-a^x\) to approach zero from below.

Step 3: Analyze the Function for \(x \geq 0\)

For \(x \geq 0\), the function \(-a^{-x}\) is also exponential, but with a positive exponent. Here, as \(x\) increases, \(a^{-x}\) becomes larger, pushing \(-a^{-x}\) sharply downward. At \(x = 0\), the function value is \(-1\).

Step 4: Determine the Behavior at the Boundary \(x = 0\)

At \(x = 0\), we need to consider the limit from either side. For \(x = 0\), the value of the function is \(-a^{0} = -1\). This is also the starting point for the behavior as \(x\) increases in the second piece of the function.

Step 5: Sketch the Graph

Start with \(-a^x\) for \(x < 0\). Plot points by calculating values for negative \(x\). The curve is downward sloping, approaching zero as \(x\) becomes more negative. For \(x \geq 0\), plot \(-a^{-x}\). This part begins at \(-1\) and moves sharply down as \(x\) increases. The graph has a discontinuity at \(x=0\), remaining distinct left and right sides.

Key Concepts

Exponential Functions

An exponential function is one where a constant base is raised to a variable power. This base can be any number greater than zero and not equal to one. In our function, we deal with the base as a constant, denoted as \(a\), and this base is set between 0 and 1. This choice of base affects the shape and direction of the graph significantly.

For the given piecewise function, two exponentials are considered:

• \(-a^x\) for \(x < 0\)
• \(-a^{-x}\) for \(x \geq 0\)

When \(x < 0\), the exponential \(-a^x\) grows smaller as \(x\) decreases since \(a\) is less than 1; thus \(a^x\) approaches zero, leading \(-a^x\) to get closer to zero from a negative perspective. Meanwhile, for \(x \geq 0\), \(-a^{-x}\) increases rapidly in magnitude negatively, indicating steep descent in the function's graph. Exponential functions often have a key characteristic: they scale the same way regardless of the function's span, leading to very steep or shallow trajectories.

Graphing Functions

Graphing functions, especially piecewise functions like this one, requires special attention to the behavior of the function in its different segments. Here, we consider the two separate parts of the piecewise function:

• For \(x < 0\), you plot points using the expression \(-a^x\), ending up with a downward sloping graph as it approaches zero.
• For \(x \geq 0\), you plot using \(-a^{-x}\), resulting in a curve that starts at \(-1\) and moves downward steeply.

To begin graphing, select points for both regions of \(x\) and substitute them into the function to get corresponding \(y\) values. Connect these points smoothly to reflect the exponential nature of the curve. Keep in mind that the transition point—where the piecewise parts change—needs extra care for accurate portrayal.

Continuity and Discontinuity

Continuity in a function graph means that the graph is a single unbroken curve. If there are breaks or gaps, we call those places discontinuities. In this piecewise function, a discontinuity occurs at \(x = 0\). This is because the expressions \(-a^{x}\) and \(-a^{-x}\) do not meet at a single, seamless point.

• Both pieces meet at a value of \(-1\) at \(x = 0\), yet they stem from different sides of the function, resulting in a break or jump in the graph.
• Because the left-hand and right-hand limits as \(x\) approaches 0 do not coincide with the function's actual value at \(x = 0\), there is no continuous curve across this point.

Understanding where discontinuities exist in a piecewise or any function is crucial as it impacts how we interpret the behavior and properties of the entire function graph.