Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=|x|\) and \(g(x)=1 / x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x) \cdot g(x)\). Where is \(h(x)\) unde ned? Note: You must deal with the cases \(x>0\) and \(x<0\) separately. This is standard protocol for handling absolute values.

Step-by-Step Solution

Answer

The respective graphs of functions \(f(x) = |x|\) and \(g(x) = 1 / x\) when combined yield the graph function \(h(x) = f(x) \cdot g(x)\), which is a horizontal line at \(y = 1\) with a hole at \(x = 0\). \(h(x)\) is undefined at \(x = 0\).

Step 1: Graphing the Function \(f(x) = |x|\)

Since \(f(x) = |x|\), the absolute value, the resulting graph will be a V-shaped graph that intersects the y-axis at (0,0) and is symmetric about the y-axis. For \(x>0\), \(y = x\), and for \(x<0\), \(y = -x\).

Step 2: Graphing the Function \(g(x) = 1 / x\)

The graph of \(g(x) = 1 / x\) is a hyperbola along the axes, with the function undefined at \(x = 0\). For \(x>0\), the graph lies in the first quadrant and gradually gets closer to the x and y-axes. For \(x<0\) it lies in the third quadrant with similar behavior.

Step 3: Combining to get \(h(x)\)

For \(x>0\), \(h(x) = x \cdot (1 / x) = 1\), it becomes a horizontal line segment at \(y = 1\) for \(x > 0\). And, for \(x<0\), \(h(x) = -x \cdot (-1 / x) = 1\), it becomes a horizontal line segment at \(y = 1\) but for \(x < 0\).

Step 4: Identify where \(h(x)\) is undefined

Since the graph of \(h(x)\) is undefined at \(x = 0\), there is a hole at \(x = 0\).

Key Concepts

Understanding the Absolute Value Function

The absolute value function, noted as \(f(x) = |x|\), is one of the foundational functions in mathematics. It essentially takes any real number and returns its non-negative value. For instance, \(|3| = 3\) and \(|-3| = 3\). The absolute value function creates a V-shaped graph. The vertex of this V is at the origin, \((0,0)\), reflecting the point where the input value equals zero.

• For values of \(x > 0\), the function outputs the same value as the input, so it rests along the line \(y = x\).
• For values of \(x < 0\), it outputs the positive version of the input, taking the shape \(y = -x\) which mirrors the positive side.

The symmetry about the y-axis is a key characteristic of this function, making it easy to visualize and graph. Its simplicity and transformation properties make it a useful tool in graphing more complex functions.

Breaking Down Rational Functions

Rational functions, expressed generally as \(g(x) = \frac{1}{x}\), involve ratios of polynomials. In our exercise, the function \(g(x) = \frac{1}{x}\) leads to a distinct hyperbola shape on the Cartesian plane. This graph is interesting because it breaks into two curves corresponding to the sectors where \(x > 0\) and \(x < 0\).

• For values of \(x > 0\), the graph descends in the first quadrant, approaching but never touching the axes, thereby showing asymptotic behavior.
• When \(x < 0\), the graph is similar but placed in the third quadrant, maintaining the same asymptotic tendencies.

One critical aspect of rational functions is understanding where they become undefined, which typically occurs where the denominator equals zero. For \(g(x) = \frac{1}{x}\), this undefined point is simply \(x = 0\). Recognizing these discontinuities is vital when working with compositions involving rational functions.

Graphing Composed Functions

Graphing a composed function like \(h(x) = f(x) \cdot g(x)\) requires understanding the behavior of its constituent parts. In this case, we combine \(|x|\) and \(\frac{1}{x}\). The resultant function, \(h(x) = |x| \cdot \frac{1}{x}\), simplifies to a constant function away from the origin.

• For \(x > 0\), since \(|x| = x\) and \(\frac{1}{x}\) yields \(1\), \(h(x)\) charts a horizontal line at \(y = 1\).
• For \(x < 0\), \(|x| = -x\) and \(\frac{-1}{x}\) still results in \(1\), forming a similar horizontal line.

Importantly, \(h(x)\) is undefined at \(x = 0\) due to \(g(x)'s\) undefined nature, creating a noticeable gap or hole in the graph at this point.
Understanding these characteristics provides insight into how function behavior is influenced by components with different properties, such as the symmetry of absolute values and the asymptotes of rational functions.