Problem 39 - Calculus with Concepts in Calculus

Question

Determine which of the following functions are even, which are odd, and which are neither. a. \(f(x)=-x\) b. \(f(x)=5 x^{2}-3\) c. \(f(x)=x^{3}+1\) d. \(f(x)=(x-2)^{2}\) e. \(f(x)=\left(x^{2}+3\right)^{3}\) f. \(y=x\left(x^{2}+1\right)^{2}\) g. \(y=\frac{x}{x^{2}+4}\) h. \(y=|x|\) i. \(y=\frac{|x|}{x}\)

Step-by-Step Solution

Answer

a) Odd, b) Even, c) Neither, d) Neither, e) Even, f) Odd, g) Odd, h) Even, i) Odd.

Step 1: Understand Even and Odd Functions

An even function satisfies the condition \(f(-x) = f(x)\). An odd function satisfies the condition \(f(-x) = -f(x)\). If neither condition is met, the function is neither even nor odd.

Step 2: Check Function (a)

Function is \(f(x) = -x\). Calculate \(f(-x) = -(-x) = x\). Since \(f(-x) eq f(x)\) and \(f(-x) = -f(x)\), the function is odd.

Step 3: Check Function (b)

Function is \(f(x) = 5x^2 - 3\). Calculate \(f(-x) = 5(-x)^2 - 3 = 5x^2 - 3\). Since \(f(-x) = f(x)\), the function is even.

Step 4: Check Function (c)

Function is \(f(x) = x^3 + 1\). Calculate \(f(-x) = (-x)^3 + 1 = -x^3 + 1\). Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), the function is neither even nor odd.

Step 5: Check Function (d)

Function is \(f(x) = (x-2)^2\). Calculate \(f(-x) = (-x-2)^2 = (x+2)^2\). Since \(f(-x) eq f(x)\), the function is neither even nor odd.

Step 6: Check Function (e)

Function is \(f(x) = (x^2 + 3)^3\). Calculate \(f(-x) = ((-x)^2 + 3)^3 = (x^2 + 3)^3\). Since \(f(-x) = f(x)\), the function is even.

Step 7: Check Function (f)

Function is \(y = x(x^2 + 1)^2\). Calculate \(y(-x) = (-x)((-x)^2 + 1)^2 = -x(x^2 + 1)^2\). Since \(f(-x) = -f(x)\), the function is odd.

Step 8: Check Function (g)

Function is \(y = \frac{x}{x^2 + 4}\). Calculate \(y(-x) = \frac{-x}{x^2 + 4}\). Since \(f(-x) = -f(x)\), the function is odd.

Step 9: Check Function (h)

Function is \(y = |x|\). Calculate \(y(-x) = |-x| = |x|\). Since \(f(-x) = f(x)\), the function is even.

Step 10: Check Function (i)

Function is \(y = \frac{|x|}{x}\). Calculate \(y(-x) = \frac{|-x|}{-x} = -\frac{|x|}{x}\). Since \(f(-x) = -f(x)\), the function is odd.

Key Concepts

Even Functions

Even functions play a special role in mathematics due to their symmetric properties. An even function is characterized by the rule that for every input \(x\), the output produced by the function when \(x\) is replaced with \(-x\) remains unchanged. In mathematical terms, this is written as \(f(-x) = f(x)\).

The beauty of even functions lies in their symmetry. When you graph an even function, you will notice that the curve is symmetrical about the y-axis. This means if you fold the graph along the y-axis, both halves would match exactly. Some classic examples of even functions include \(f(x) = x^2\) and \(g(x) = \cos(x)\).

To identify an even function from a given expression, substitute \(-x\) in place of \(x\) and simplify. If the resulting expression is the same as the original function, then it's even. For example, with \(h(x) = 5x^2 - 3\), substituting \(-x\) gives \(5(-x)^2 - 3 = 5x^2 - 3\), confirming it's an even function.

Odd Functions

Odd functions are another intriguing category known for their unique symmetry properties. An odd function satisfies the condition \(f(-x) = -f(x)\) for all \(x\), meaning that replacing \(x\) with \(-x\) and then taking the negative of the entire function gives you back the original function value.

Graphically, odd functions exhibit symmetry about the origin. Imagine rotating the graph 180 degrees around the origin (the point where the x-axis and y-axis cross); if the graph looks identical in its new position, then the function is odd. Popular examples include \(f(x) = x^3\) and \(g(x) = \sin(x)\).

To test if a function is odd, replace \(x\) with \(-x\) in the function and simplify. If simplification leads to a negative version of the original function, it's odd. For instance, taking \(y = \frac{x}{x^2 + 4}\), substituting \(-x\) results in \(\frac{-x}{x^2 + 4}\), which is the negative of the original function, confirming it's odd.

Function Symmetry

Function symmetry is a powerful tool in understanding the behavior of functions. Symmetry can tell us a lot about the function's graph without needing to plot every point.

There are mainly two types of symmetries to check in functions: symmetry about the y-axis and symmetry about the origin.

• Symmetry about the y-axis: If a function is symmetrical across the y-axis, changing \(x\) to \(-x\) will not affect the output, indicating an even function.
• Symmetry about the origin: When a function has this symmetry, flipping the graph both vertically and horizontally results in the same graph, which defines an odd function.

Understanding these symmetries helps in sketching functions faster and predicting their behavior in applied scenarios. By checking these symmetries, students can easily classify functions and verify if they are even or odd. This provides a basis for deeper analysis in higher mathematics.