Understanding Even, Odd, and Neither Functions

The video tutorial explains how to determine if a function is even, odd, or neither. It begins with a review of the definitions and graphical characteristics of even and odd functions. The instructor then provides examples, starting with an even function, demonstrating both graphically and algebraically why it is even. The next example shows a function that is neither even nor odd, with a focus on graphical analysis. Finally, the video concludes with an algebraic verification that the function is neither even nor odd, reinforcing the concepts discussed.

What is the condition for a function to be classified as even?

• f(x) = f(-x)

• f(x) = x

• f(x) = -f(x)

• f(x) = 0

Which type of symmetry does an odd function's graph exhibit?

• Symmetry across the y-axis

• Rotational symmetry about the origin

• No symmetry

• Symmetry across the x-axis

What is the first step in determining if a function is even, odd, or neither?

• Integrate the function

• Check the graph of the function

• Solve the function algebraically

• Find the derivative

In the first example, what was the algebraic expression for f(-x)?

• 2/(-x)^2 - 4

• 2/x^2 - 4

• 2x^2 - 4

• 2/x - 4

What was the conclusion about the first example function after algebraic verification?

• The function is odd

• The function is even

• The function is undefined

• The function is neither

What was observed about the graph of the second example function?

• It has rotational symmetry about the origin

• It is neither symmetrical across the y-axis nor has rotational symmetry

• It is symmetrical across the x-axis

• It is symmetrical across the y-axis

What is the algebraic expression for f(-x) in the second example?

• 1/4x^3 + x^2 - x - 3

• -1/4x^3 + x^2 + x - 3

• -1/4x^3 - x^2 - x + 3

• 1/4x^3 - x^2 + x + 3

What was the conclusion about the second example function after algebraic verification?

• The function is both even and odd

• The function is neither

• The function is odd

• The function is even

What must be shown to prove a function is not even?

• f(x) = -f(x)

• f(x) = f(-x)

• f(x) ≠ f(-x)

• f(x) ≠ -f(x)

What must be shown to prove a function is not odd?

• -f(x) ≠ f(-x)

• -f(x) = f(-x)

• f(x) ≠ f(-x)

• f(x) = f(-x)