Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(G(x)=[2 x-1]\)

Step-by-Step Solution

Answer

The function \(G(x)\) is neither even nor odd.

Step 1: Understand the Definitions

A function is **even** if for all values of \(x\), \(f(-x) = f(x)\). It is **odd** if \(f(-x) = -f(x)\). If neither condition is satisfied, the function is categorized as **neither**.

Step 2: Substitute and Simplify for Evenness

To check if \(G(x) = [2x - 1]\) is even, substitute \(-x\) into the function: \(G(-x) = [2(-x) - 1] = [-2x - 1]\). Check if \(G(-x) = G(x)\): \([-2x - 1]\) is not equal to \([2x - 1]\). Hence, \(G(x)\) is not even.

Step 3: Substitute and Simplify for Oddness

Now, to check if it is odd, see if \(G(-x) = -G(x)\). Calculating \(-G(x)\): \(-G(x) = -[2x - 1] = [-2x + 1]\). Since \(G(-x) = [-2x - 1]\), and it does not equal \([-2x + 1]\), thus \(G(-x) eq -G(x)\). So, \(G(x)\) is not odd.

Step 4: Conclusion on Evenness or Oddness

Since \(G(x)\) is not even and not odd, it is categorized as **neither**.

Step 5: Sketch the Graph

The function \(G(x) = [2x - 1]\) involves the greatest integer less than or equal to \(2x - 1\). Sketching this involves creating a step graph where each integer value corresponds to a piecewise constant segment. Evaluate points to see the changes, such as at \(x = 0.5\), \(x = 1\), and so forth, transitioning at every half interval.

Key Concepts

Greatest Integer Function

The greatest integer function is a unique mathematical function that identifies the largest integer less than or equal to a given number. When this function is denoted as \[ \lfloor x \rfloor \], it simply "steps down" to the nearest whole number beneath the original value. Whether you have \([2.3]\), \([-1.7]\), or any other non-integer, the greatest integer function transforms the input into a more 'number-friendly' integer value by truncating the decimal part.

The significance of this function lies in its role within step graph creation – it provides a neat way to explore piecewise constant functions. Functions such as \[ G(x) = [2x - 1] \] utilize the greatest integer function to create the stepping effect, defining graph behavior across different ranges based on its integer outputs.

Function Symmetry

Function symmetry helps determine if a function is either even or odd, both of which are vital for understanding graph behaviors. An even function exhibits symmetry across the y-axis. This means wherever you have the input \(x\), the output remains the same even if you substitute \(-x\). In contrast, odd functions have rotational symmetry around the origin. Thus, flipping \(x\to -x\) results in a direct reflection of the function over the origin, maintaining negative parity.

To determine whether \(G(x) = [2x - 1]\) is even or odd, you revert to these tests. Not all functions perfectly fit these criteria; as with \(G(x)\), neither symmetry condition held true. Hence, it is labeled as neither even nor odd.

Step Graph

A step graph results when you plot a greatest integer function or similar functions that maintain constant y-values over specific intervals. These graphs are distinguishable by their characteristic horizontal line segments—"steps"—interrupted by abrupt vertical jumps.

In the case of \[ G(x) = [2x - 1] \], the graph builds by evaluating how the function behaves at distinct points. As you plot it, each interval between integer input values results in a horizontal segment, shifting "up" or "down" only when crossing an integer threshold. This stepping is illustrated prominently when observing the function's behavior across transitions like around \([0.5, 1]\) and similar half-intervals.

Understanding how step graphs work is fundamental for recognizing patterns within integer-based functions, revealing symmetry, periodicity, and discontinuities visually and effectively.