Problem 79 - 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 \(x\). $$f(t)=t^{2}+2 t-3$$

Step-by-Step Solution

Answer

The function \(f(t)=t^{2}+2 t-3\) is neither an even function nor an odd function.

Step 1: Algebraic Analysis

Replace \(t\) in the function \(f(t)=t^{2}+2 t-3\) with \(-t\), we obtain \(f(-t) = (-t)^2 - 2t - 3 = t^2 - 2t - 3\) which differs from the original function \(f(t)\) and is not a negation of \(f(t)\) either. Hence, the function is neither even nor odd algebraically.

Step 2: Graphical Analysis

Plot the function \(f(t)=t^{2}+2 t-3\) using a graphing utility. The shape of the graph is a parabola and it opens upwards, it does not possess any symmetry about the y-axis or origin. Therefore, the function is neither even nor odd graphically.

Step 3: Numerical Analysis

Using a graphing utility, one would set up a table to list out values of \(f(x)\) and \(f(-x)\) for a range of \(x\) values. However, as it has already been determined algebraically and graphically that the function is neither even nor odd, it would also hold true numerically.

Key Concepts

Algebraic Analysis

To determine if a function is even or odd algebraically, we perform some substitutions. An even function has the property that replacing \(x\) with \(-x\) doesn't change the function: \(f(x) = f(-x)\). For an odd function, replacing \(x\) with \(-x\) results in a negation: \(f(-x) = -f(x)\). In our example, we substitute \(t\) in \(f(t) = t^2 + 2t - 3\) with \(-t\), resulting in:

• \(f(-t) = (-t)^2 + 2(-t) - 3 = t^2 - 2t - 3\)

This changes the equation from the original, and is also not its negation. Hence, the function doesn't fit the criteria for being even or odd. It's crucial in algebraic analysis to use these definitions for determining the function's symmetry.

Graphical Analysis

Graphical analysis involves visualizing the function's graph to examine its symmetry. An even function is symmetric about the y-axis, while an odd function is symmetric about the origin. Utilizing a graphing utility, plot \(f(t) = t^2 + 2t - 3\). The graph forms a parabola that opens upwards:

• Even symmetry would reflect the right side over the y-axis, which is not present.
• Odd symmetry would involve a half-turn around the origin which is also not present.

Therefore, visually, the function shows neither even nor odd symmetry. Graphical analysis helps provide a quick visual confirmation of algebraic findings.

Numerical Analysis

Numerical analysis uses calculated values to verify whether a function is even, odd, or neither. Using a graphing utility’s table feature, calculate values for \(f(x)\) and \(f(-x)\) over a set of \(x\) values. For instance, consider values like \(x = 1\), \(x = -1\), and various other numbers.

• If \(f(x) = f(-x)\) for all chosen values, the function is even.
• If \(f(-x) = -f(x)\), it’s odd.

However, this function doesn’t satisfy either condition across different values, confirming once again it's neither. Numerical analysis provides concrete data to support algebraic and graphical insights.