Problem 97 - Calculus Volume 1

Question

For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. $$ g(x)=\left\\{\begin{array}{l}{\frac{3}{x-2}, x \neq 2} \\ {4, x=2}\end{array} ; g(0) ; g(-4) ; g(2)\right. $$

Step-by-Step Solution

Answer

g(0) = -\frac{3}{2}, g(-4) = -\frac{1}{2}, g(2) = 4.

Step 1: Evaluate g(0)

Since 0 is not equal to 2, we use the rule for \( x eq 2 \). Substitute \( x = 0 \) into the function: \( g(0) = \frac{3}{0 - 2} = \frac{3}{-2} = -\frac{3}{2} \).

Step 2: Evaluate g(-4)

Since -4 is not equal to 2, we again use the rule for \( x eq 2 \). Substitute \( x = -4 \) into the function: \( g(-4) = \frac{3}{-4 - 2} = \frac{3}{-6} = -\frac{1}{2} \).

Step 3: Evaluate g(2)

Since \( x = 2 \), we use the rule specifically for \( x = 2 \), which gives us \( g(2) = 4 \).

Step 4: Sketch the Graph

The graph consists of two parts: for \( x eq 2 \), plot the hyperbola \( y = \frac{3}{x-2} \). For \( x = 2 \), plot a point at (2, 4). Note that at \( x = 2 \), there is a discontinuity, represented by the point (2, 4), distinct from the hyperbolic curve.

Key Concepts

Evaluating Functions

Evaluating functions is an important skill when working with piecewise-defined functions. It involves substituting a specific value of the independent variable (usually denoted as \( x \)) into the function to determine the corresponding value of the dependent variable (usually \( y \) or \( g(x) \) in this case).

For the function given in the exercise:

• When evaluating \( g(0) \), we determine which rule of the piecewise function applies. Since 0 is not equal to 2, we use the first rule: \( g(x) = \frac{3}{x-2} \). Substituting 0 gives: \( g(0) = \frac{3}{0-2} = -\frac{3}{2} \).
• For \( g(-4) \), again, \( x \) is not 2, so we apply the same rule: \( g(-4) = \frac{3}{-4-2} = -\frac{1}{2} \).
• For \( g(2) \), since \( x = 2 \), we use the second rule where \( g(x) = 4 \).

Understanding how to correctly choose which piece of the function to use is crucial for accurate evaluation.

Graphing Functions

Graphing piecewise functions involves plotting each segment according to its rule over its applicable domain. In our example, the function has two distinct rules:

• The first part is a hyperbola given by \( y = \frac{3}{x-2} \) for all \( x eq 2 \). This means the graph will look like a curve that approaches but never touches the vertical line \( x = 2 \).
• The second part is simply the point \((2, 4)\) when \( x = 2 \). This point is isolated because the rule \( y = \frac{3}{x-2} \) does not apply at \( x = 2 \).

When sketching piecewise functions, always plot each segment carefully, respecting its domain. Make sure to clearly indicate any distinct points that do not align with the continuous sections of the function.

Discontinuity in Graphs

Discontinuities in graphs are spots where the function is not continuous, meaning the graph of the function has jumps, breaks, or holes. In piecewise functions, these discontinuities often occur at the points where the rules change.

• In our function, there is a discontinuity at \( x = 2 \). The graph of \( y = \frac{3}{x-2} \) cannot be plotted at \( x = 2 \) because division by zero is undefined, thus creating a break in the curve of the graph.
• Instead, the function at \( x = 2 \) is specifically defined by a distinct point \( (2, 4) \), which does not coincide with the hyperbolic curve.

Recognizing discontinuities is essential because it affects the function's graph and also highlights why certain limits may not exist or why the function lacks continuity at specific points.