Problem 109 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It Let \(f\) be an even function. Determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=-f(x+3)\)

Step-by-Step Solution

Answer

(a) \(g(x) = -f(x)\) is odd. (b) \(g(x) = f(-x)\) is even. (c) \(g(x) = f(x) - 2\) is even. (d) \(g(x) = -f(x + 3)\) is neither odd nor even.

Step 1: Determine the nature of function g(x) = -f(x)

Substitute \(x\) with \(-x\) in the function \(g(x) = -f(x)\), thus we get \(g(-x) = -f(-x)\). However, since \(f\) is even, \(f(-x) = f(x)\) so, \(g(-x) = -f(x) = -g(x)\). Therefore, \(g(x)\) is an odd function.

Step 2: Determine the nature of function g(x) = f(-x)

Substitute \(x\) with \(-x\) in the function \(g(x) = f(-x)\), thus we get \(g(-x) = f(x)\). Since \(g(-x) = g(x)\), \(g(x)\) is an even function.

Step 3: Determine the nature of function g(x) = f(x) - 2

Substitute \(x\) with \(-x\) in the function \(g(x) = f(x) - 2\), thus we get \(g(-x) = f(-x) - 2 = f(x) - 2 = g(x)\). Therefore, \(g(x)\) is an even function.

Step 4: Determine the nature of function g(x) = -f(x+3)

Substitute \(x\) with \(-x\) in the function \(g(x) = -f(x+3)\), then we get \(g(-x) = -f(-x+3) = -f(x+3) ≠ g(x)\). Thus, \(g(x)\) is neither, since it does not obey the rules of either an odd or even function.

Key Concepts

Function Transformation

Understanding function transformations is essential in determining how functions shift, stretch, rotate, or flip on a graph. Function transformation refers to the modification of a function's attributes, like its domain, range, and overall graph shape. These transformations can occur in several ways, each governed by specific operations:

• Translation: Moves the function horizontally or vertically.
• Reflection: Flips the function over a specific axis, such as the x-axis or y-axis.
• Scaling: Stretches or compresses the function's graph.

In the case of this exercise, transformations help identify the nature of \( g(x) = -f(x) \), \( g(x) = f(-x) \), \( g(x) = f(x) - 2 \), and \( g(x) = -f(x+3) \). By applying transformations and checking their effect on the graph and symmetry of the function, we can ascertain whether these new functions are even, odd, or neither.

Even Function

An even function is characterized by its symmetry about the y-axis. This means that for every input \(x\), the function satisfies the condition:

• \(f(-x) = f(x)\)

Even functions like the cosine function, \(f(x) = \cos{x}\), maintain the same value if you replace \(x\) with \(-x\). In this exercise,

• \(g(x) = f(-x)\) remains even because replacing \(x\) with \(-x\) yields \(g(-x) = f(x)\) which equals \(g(x)\).
• Similarly, \(g(x) = f(x) - 2\) is also even as it retains the symmetry of \(f(x)\) about the y-axis after a vertical translation of 2 units.

These functions maintain the criteria of the original even function when transformed in these specific manners.

Odd Function

Odd functions possess a distinctive point symmetry about the origin, implying they exhibit a rotational symmetry of 180 degrees around the origin. The mathematical characteristic of an odd function is:

• \(f(-x) = -f(x)\)

These functions, such as sine \(f(x) = \sin{x}\), have outputs that are the negative of the original when \(x\) is replaced by \(-x\). From the exercise, we learn that:

• \(g(x) = -f(x)\) transforms from an even function to an odd function. When you substitute \(x\), the resulting \(g(-x) = -f(-x) = -f(x)\) indeed gives \(-g(x)\), confirming \(g(x)\) as an odd function.

This transformation involves a reflection over the x-axis, switching the even function's property to that of an odd function.

Function Properties

Functions possess specific properties that allow us to classify and analyze them based on symmetry, continuity, and transformations. Understanding these properties clarifies whether a function is even, odd, or neither:

• An even function complements itself on opposite sides of the y-axis.
• An odd function complements itself across the origin, such that their graphs rotate 180 degrees around it.
• Functions like \(g(x) = -f(x+3)\), which lack symmetry, are neither. The combination of horizontal translation and reflection over the x-axis breaks the symmetrical pattern needed to retain even or odd properties.

By evaluating transformations like reflections and translations, we can predict the resultant function's behavior and properties, which is crucial in calculus and other advanced mathematical applications.