Problem 13 - The Calculus with Analytic Geometry

Question

For each of the following functions, determine whether \(f\) is even, odd, or neither. (a) \(f(x)=2 x^{4}-3 x^{2}+1\) (b) \(f(x)=5 x^{3}-7 x\) (c) \(f(s)=s^{2}+2 s+2\) (d) \(f(x)=x^{6}-1\) (e) \(f(t)=5 t^{7}+1\) (f) \(f(x)=|x|\) (g) \(f(y)=\frac{y^{3}-y}{y^{2}+1}\) (h) \(f(x)=\frac{x-1}{x+1}\)

Step-by-Step Solution

Answer

(a) Even (b) Odd (c) Neither (d) Even (e) Neither (f) Even (g) Odd (h) Neither

Step 1: Definition of Even, Odd, and Neither Functions

A function is even if for all values of x, f(-x) = f(x). A function is odd if for all values of x, f(-x) = -f(x). If neither condition holds, the function is neither even nor odd.

Step 2: Check Function (a)

For \(f(x)=2 x^{4}-3 x^{2}+1\), calculate \(f(-x)\):\(f(-x) = 2(-x)^4 - 3(-x)^2 + 1 = 2x^4 - 3x^2 + 1\). Since \(f(-x) = f(x)\), the function is even.

Step 3: Check Function (b)

For \(f(x)=5 x^{3}-7 x\), calculate \(f(-x)\):\(f(-x) = 5(-x)^3 - 7(-x) = -5x^3 + 7x = - (5x^3 - 7x) = - f(x)\). Since \(f(-x) = -f(x)\), the function is odd.

Step 4: Check Function (c)

For \(f(s)=s^{2}+2s+2\), calculate \(f(-s)\):\(f(-s) = (-s)^2 + 2(-s) + 2 = s^2 - 2s + 2\). Since \(f(-s) eq f(s)\) and \(f(-s) eq -f(s)\), the function is neither even nor odd.

Step 5: Check Function (d)

For \(f(x)=x^{6}-1\), calculate \(f(-x)\):\(f(-x) = (-x)^6 - 1 = x^6 - 1\). Since \(f(-x) = f(x)\), the function is even.

Step 6: Check Function (e)

For \(f(t)=5 t^{7}+1\), calculate \(f(-t)\):\(f(-t) = 5(-t)^7 + 1 = -5t^7 + 1\). Since \(f(-t) eq f(t)\) and \(f(-t) eq -f(t)\), the function is neither even nor odd.

Step 7: Check Function (f)

For \(f(x)=|x|\), calculate \(f(-x)\):\(f(-x) = |-x| = |x| = f(x)\). Since \(f(-x) = f(x)\), the function is even.

Step 8: Check Function (g)

For \(f(y)=\frac{y^{3}-y}{y^{2}+1}\), calculate \(f(-y)\): \(f(-y) = \frac{(-y)^3 - (-y)}{(-y)^2 + 1} = \frac{-y^3 + y}{y^2 + 1} = -\frac{y^3 - y}{y^2 + 1} = -f(y)\). Since \(f(-y) = -f(y)\), the function is odd.

Step 9: Check Function (h)

For \(f(x)=\frac{x-1}{x+1}\), calculate \(f(-x)\): \(f(-x) = \frac{-x-1}{-x+1} = \frac{-(x+1)}{-(x-1)} = \frac{x+1}{x-1}\). Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), the function is neither even nor odd.

Key Concepts

Function Analysis

Understanding if a function is even, odd, or neither begins with the analysis of the function's definition. This means checking how substituting \(-x\) for \(x\) affects the function. For a function \(f(x)\) to be *even*, it must satisfy the condition \ f(-x) = f(x) \. This symmetry means that the function is mirrored over the y-axis. In contrast, for a function to be *odd*, it must satisfy \ f(-x) = -f(x) \. Here, the function is symmetric about the origin. To sum up function analysis:

• *Identify* the function type by substituting -x.
• *Evaluate* whether f(-x) equals f(x) for even, -f(x) for odd, or neither.

This structured approach helps break down complex calculations and identify symmetries effortlessly.

Symmetry in Functions

Symmetry is a key concept in understanding whether a function is even or odd.
For *even functions*, the symmetry is about the y-axis, meaning if you fold the graph along the y-axis, both halves will match.
For example, consider the function \(f(x) = 2x^4 - 3x^2 + 1\). Calculating \(f(-x)\) as shown in the original solution, we get \ f(-x) = 2x^4 - 3x^2 + 1 \, which is identical to \ f(x) \, confirming that this function is even.
On the other hand, *odd functions* exhibit rotational symmetry around the origin. This means if you rotate the graph 180 degrees around the origin, it remains unchanged.
Take the function \(f(x) = 5x^3 - 7x\). By calculating \ f(-x) = -5x^3 + 7x = - f(x) \ , as shown in the solution, we see that \(f(-x) \) is the negative of \(f(x)\), thus confirming its odd nature.
Summarizing, symmetry helps us visualize and understand the properties of functions more clearly.

Calculus Basics

Grasping the basics of calculus involves understanding function behaviors and their different properties. Here, the focus is on even and odd functions.
Even functions, described as symmetrical about the y-axis, will have only even powers of x when expanded, or symmetrical terms like \(cos(x)\).
Take \(f(x) = x^6 - 1\). The higher power polynomial with even powers (6 in this case) aligns perfectly with the requirement as when calculating \ f(-x) = x^6 - 1 = f(x) \ it remains unchanged. This confirms its identity as even.
Odd functions, identified by symmetrical behavior around the origin, often contain odd powers of x or antisymmetric terms like \sin(x)\).
Consider \(f(x) = 5t^7 + 1\). When checking for \(f(-t) = -5t^7 + 1\), it neither matches \(f(t)\ nor \(-f(t)\), making the function neither even nor odd, illustrating the depth and detail calculus brings into function analysis.
With these basic principles, students can apply differentiation and integration to more deeply analyze function behaviors and their symmetry.