Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$g(x)=x^{3}-5 x$$

Step-by-Step Solution

Answer

The function \(g(x) = x^3 - 5x\) is neither even nor odd based on algebraic, graphical, and numerical determination.

Step 1: Algebraic determination

For a function to be even, it must satisfy \(f(x) = f(-x)\) for all \(x\). For a function to be odd, it must satisfy \(f(-x) = -f(x)\) for all \(x\). Substitute \(x\) with \(-x\) in \(g(x) = x^3-5x\): \[g(-x) = (-x)^3-5(-x) = -x^3 + 5x \] - which does not equate to \(g(x)\) or \(-g(x)\), therefore the function is neither even nor odd algebraically.

Step 2: Graphical Determination

Using a graphing utility to plot \(g(x) = x^3 - 5x\), the graph symmetry can be observed. Even functions have symmetry about the y-axis and odd functions have symmetry about the origin. The graph of this function shows that there is no symmetry about the y-axis nor about the origin, therefore the function is not even or odd graphically.

Step 3: Numerical Determination

Choose several values for \(x\) and compare \(f(x)\) to \(f(-x)\). An even function will have \(f(x) = f(-x)\) and an odd function will have \(f(x) = -f(-x)\) for all \(x\). After calculating \(f(x)\) and \(f(-x)\) for several \(x\) values, the results do not meet the conditions for either even or odd functions, so the function is neither even nor odd numerically.

Key Concepts

Even and Odd Functions

Understanding if a function is even, odd, or neither is essential in the realm of algebra and calculus. Even functions are symmetrical about the y-axis, meaning if you were to fold the graph along the y-axis, both sides would match perfectly. Mathematically, a function is considered even if it holds true that f(x) = f(-x) for all values of x. An example of an even function is f(x) = x^2.

On the other hand, odd functions have rotational symmetry about the origin, which is the point (0,0) on the graph. You can rotate the graph 180 degrees around the origin and the graph will look the same. Algebraically, a function is odd if f(-x) = -f(x) for every x value. For instance, f(x) = x^3 is an odd function.

However, not all functions fit neatly into these categories. In our exercise, g(x) = x^3 - 5x neither satisfied the condition of an even function nor that of an odd function. Hence, it was deemed neither even nor odd.

Graphical Symmetry

The method of determining evenness or oddness by looking at the function's graph offers a visual approach which can be quite insightful. Recall our exercise, where we used a graphing utility to analyze the function g(x).

Even Functions

Graphically, even functions like f(x) = x^2 display symmetry about the y-axis. If you drop a vertical line down the y-axis, the left and right side of the graph will mirror each other.

Odd Functions

In contrast, odd functions such as f(x) = x^3 demonstrate symmetry about the origin. Imagine turning the graph upside down, and it would still be the same, that's the signature trait of an odd function.

Neither Even Nor Odd

For g(x) = x^3 - 5x, when plotted, the graph did not showcase symmetry about the y-axis or about the origin. This absence of symmetry is a clear indication that the function does not fit the criteria for being classified as even or odd.

Algebraic Analysis

The algebraic method is a direct approach to determining the nature of a function with respect to evenness or oddness. It involves manipulating the function algebraically and comparing it against certain criteria.

In the exercise, we substituted x with -x in the equation g(x) = x^3 - 5x. The resulting function, g(-x) = -x^3 + 5x, did not satisfy the conditions for being either even (g(x) = g(-x)) or odd (g(-x) = -g(x)). This algebraic inspection provided a clear cut answer, reinforcing the conclusion made from both the graphical and numerical methods.

The algebraic analysis is a vital step in identifying function symmetry because it gives a definitive proof, without any approximation errors that might come from graphical interpretations or limited numerical data points.