Problem 59 - Precalculus: A Concise Course

Question

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$ g(x)=\cot x $$

Step-by-Step Solution

Answer

The function g(x) is odd.

Step 1: Understand the function and recall the definitions of even and odd functions

An even function is one where \(f(x) = f(-x)\). An odd function is one where \(f(x) = -f(-x)\). Neither condition is met, then the function is neither even nor odd. The function given is \(g(x) = cot(x)\).

Step 2: Check if the function is even

Substitute \(-x\) into the function. If the equation \(g(x) = g(-x)\) is valid, the function is even. The cotangent of \(-x\) is \(-cot(x)\), i.e., \(g(-x) = cot(-x) = -cot(x) \neq cot(x) = g(x)\). So the function is not even.

Step 3: Check if the function is odd

Check if the equation \(g(x) = -g(-x)\) is valid, in which case the function is odd. As derived in the previous step, \(g(-x) = -cot(x) = -g(x)\) which verifies that \(g(x) is an odd function.

Key Concepts

Cotangent Function

The cotangent function, which we denote as \(cot(x)\), is one of the six fundamental trigonometric functions. It is related to the more familiar tangent function but takes a unique angle—literally! To understand the cotangent function, consider a right triangle: cotangent is the ratio of the adjacent side to the opposite side, which is essentially the reciprocal of the tangent function.

Now, if we plot \(cot(x)\) on a Cartesian plane, we'll see a repeating pattern, but with a catch: the cotangent function has undefined points, or asymptotes, where the sine function equals zero, since cotangent is the ratio of cosine to sine. This distinguishes \(cot(x)\) from other trigonometric functions that are always defined. The function's period is \(\text{π}\), meaning that it repeats every \(\text{π}\) units along the x-axis.

Understanding the behavior of \(cot(x)\) is crucial when assessing its symmetry properties, which are intrinsic to determining if a function is even, odd, or neither. Remember, trigonometric functions, including cotangent, can often have these symmetrical properties understood through their graphs and algebraic expressions.

Algebraic Verification

When we talk about algebraic verification, we're essentially double-checking the characteristics of functions using algebraic methods. So how does this apply to our cotangent function regarding its parity—that is, whether it's odd, even, or neither? Here's the critical algebraic insight: for a function to be even, any input \(x\) should yield the same output as \(-x\), symbolically, \(f(x) = f(-x)\). For a function to be odd, the output for \(-x\) should be the negative of the output for \(x\), which is \(f(x) = -f(-x)\).

If neither of these conditions is satisfied, then the function doesn't have symmetry about the y-axis (even) or origin (odd). With the cotangent function, when we substitute \(-x\) into the function, we get \(-cot(x)\), which does not equal \(cot(x)\). This immediately tells us the function isn't even. But since \(cot(-x) = -cot(x)\), that tells us that our cotangent function is odd. Algebraic verification is a powerful tool and serves as a foundational check against what we can see on a graph.

Graph Analysis

Analyzing the graph of a function can often tell you a lot about its properties, including symmetry. When we plot our cotangent function, \(g(x) = \cot x\), we're looking for specific kinds of symmetry. For even functions, we expect the graph to be symmetrical about the y-axis. In other words, for every point on the graph, its mirror image exists on the opposite side of the y-axis.

For odd functions, the symmetry we're after is rotational symmetry about the origin. Picturing a point on the graph, if you rotate it 180 degrees around the origin, it should land on another point of the function. With \(g(x) = \cot x\), the graph doesn't mirror across the y-axis; instead, it exhibits rotational symmetry, showing us that the function is odd.

This visual confirmation through graph analysis is vital as it supports the algebraic verification. When learning trigonometry or any other mathematical concept that involves functions, taking the time to look at the graph can provide intuitive understanding and reinforce what the algebra tells us.