LE COURS : Fonction paire, fonction impaire. Fonctions de référence - Seconde

Summary

This video covers the essential concepts of even and odd functions, explaining their graphical and algebraic properties. The instructor demonstrates that even functions are symmetric about the y-axis, while odd functions are symmetric around the origin. Key examples include the square function, which is even, and the cube function, which is odd. The lesson also covers other functions such as the inverse and square root functions, explaining their parities and characteristics. The video emphasizes the importance of symmetry in understanding function properties and encourages further practice to master the concepts.

Takeaways

• 😀 Functions can be classified as even or odd based on their symmetry properties.
• 😀 An even function is symmetric with respect to the y-axis, meaning f(x) = f(-x) for all x in its domain.
• 😀 Odd functions exhibit symmetry about the origin, meaning f(x) = -f(-x) for all x in its domain.
• 😀 The square function (f(x) = x^2) is an example of an even function as it is symmetric with respect to the y-axis.
• 😀 The cube function (f(x) = x^3) is an example of an odd function because it is symmetric with respect to the origin.
• 😀 The inverse function (f(x) = 1/x) is also an odd function, exhibiting symmetry about the origin.
• 😀 The square root function (f(x) = √x) is neither even nor odd because it is not defined for negative values of x.
• 😀 When analyzing whether a function is even or odd, one can check the symmetry of the graph or use algebraic definitions based on the behavior of f(x) and f(-x).
• 😀 The graph of an even function (like the square function) is symmetric about the y-axis, while the graph of an odd function (like the cube function) is symmetric about the origin.
• 😀 To prove a function is even or odd, calculate f(-x) and compare it to f(x) (for even) or -f(x) (for odd).

Q & A

What is the definition of an even function?

An even function is a function whose graph is symmetric about the y-axis. Mathematically, it satisfies the condition f(x) = f(-x) for all x in the domain of the function.

How can you recognize an even function graphically?

Graphically, an even function has symmetry around the y-axis. If you fold the graph along the y-axis, the two halves will overlap.

What is the definition of an odd function?

An odd function is a function whose graph is symmetric about the origin. It satisfies the condition f(-x) = -f(x) for all x in the domain of the function.

How can you recognize an odd function graphically?

Graphically, an odd function has rotational symmetry around the origin. If you rotate the graph by 180 degrees around the origin, it will coincide with itself.

Can you give an example of an even function?

Yes, the quadratic function f(x) = x² is an even function because its graph is a parabola that is symmetric about the y-axis.

Can you give an example of an odd function?

Yes, the cubic function f(x) = x³ is an odd function because its graph is symmetric about the origin and satisfies the property f(-x) = -f(x).

Why does the square root function f(x) = √x not qualify as an even or odd function?

The square root function is neither even nor odd because it is only defined for non-negative values of x, and it doesn't exhibit the symmetry required for even or odd functions.

What is the behavior of the inverse function f(x) = 1/x with respect to symmetry?

The inverse function f(x) = 1/x is an odd function, as its graph is symmetric about the origin. It satisfies the property f(-x) = -f(x).

How does the graph of the quadratic function behave?

The graph of the quadratic function f(x) = x² is a parabola that opens upwards and is symmetric about the y-axis, passing through the origin.

What is a key characteristic of the cubic function graph?

The cubic function graph is symmetric about the origin and has a point of inflection at the origin. It increases rapidly for both positive and negative values of x.