Exploring Even and Odd Functions

This video tutorial covers the concepts of even and odd functions, focusing on their symmetry properties. It explains how to determine if a function is even, odd, or neither using both graphical and algebraic methods. The tutorial includes examples and practice problems to reinforce understanding, highlighting the importance of symmetry in function analysis.

What is the defining characteristic of an even function?

• Symmetry with respect to the x-axis

• Symmetry with respect to the y-axis

• Symmetry with respect to the origin

• No symmetry

Which of the following is true for an odd function?

• f(x) = f(-x)

• f(-x) = f(x)

• f(-x) = -f(x)

• f(x) = -f(-x)

What happens when negative x is raised to an even power?

• The result is negative

• The result is undefined

• The result is positive

• The result is zero

What is the result of raising negative x to the third power?

• Positive x cubed

• Negative x cubed

• Undefined

• Zero

If f(-x) is not equal to f(x) and not equal to -f(x), what type of function is it?

• Odd

• Neither

• Both

• Even

For the function f(x) = x^2 - 8x - 14, what is f(-x)?

• x^2 - 8x + 14

• x^2 + 8x - 14

• -x^2 - 8x - 14

• -x^2 + 8x + 14

What is the algebraic form of an even function?

• f(x) = -f(-x)

• f(-x) = -f(x)

• f(x) = f(-x)

• f(-x) = f(x)

What is the algebraic form of an odd function?

• f(x) = f(-x)

• f(x) = -f(-x)

• f(-x) = -f(x)

• f(-x) = f(x)

What is the result of f(-x) for the function f(x) = cos(x)?

• cos(x)

• -cos(x)

• sin(x)

• -sin(x)

For the function f(x) = -√(2x^2 - x^4), what is f(-x)?

• √(2x^2 - x^4)

• -√(2x^2 - x^4)

• √(2x^2 + x^4)

• -√(2x^2 + x^4)