Question

\(f\) is an even function, continuous on \([-3,3],\) and satisfies the following. (d) What can you conclude about \(f(3)\) and \(f(-3) ?\)

Short answer

\(f(3)\) and \(f(-3)\) are equal as \(f\) is an even function.

Step-by-step solution

Understanding Even Functions

An even function is a function that satisfies \(f(x) = f(-x)\) for any number \(x\) in the function's domain. This implies that the function is symmetric with respect to the y-axis. It means that whatever the output is when \(x\) takes a particular value, it will be the same for its negative.

Apply the Even Function Property

With knowledge of the property of even functions, we can understand that if \(f\) is an even function, then \(f(3) = f(-3)\). This is because the output is the same for any input and its negative.

Key concepts

Even Function Properties

In mathematics, an even function has specific characteristics that make it easily recognizable and provide convenient properties for analysis. When dealing with even functions, you can expect the function to exhibit the following:

• The defining property of an even function is that for every number in the function's domain, we have f(x) = f(-x). This essentially means if you plug a positive value into the function or its negative counterpart, the result will be the same.
• Graphically, an even function is symmetrical about the y-axis. If you were to fold the graph along this axis, both halves would match perfectly.
• Even functions have real-world applications where symmetrical behavior is observed, such as in engineering and physics.
• Arithmetic with even functions can sometimes be simplified, especially when integrating, since the area under the curve is duplicated on both sides of the y-axis.

For example, the function f(x) = x2 is even because f(2) = 4 and f(-2) = 4, showing the equal outputs for opposite inputs. These properties help in making predictions about function values and provide a framework for mathematical arguments and proofs.

Function Symmetry

Symmetry in mathematics often refers to the balanced and proportional similarity that a shape, function, or equation exhibits. When we say a function is symmetric, we particularly mean it is either symmetric with respect to the y-axis, the x-axis, or the origin.

• For an even function, symmetry about the y-axis is the characteristic form of symmetry. This means that the left and right sides of the graph are mirror images.
• f(x) can be replaced with f(-x) without changing the function's value, which is a consequence of this symmetry.
• Symmetry can be a powerful tool in calculus, for instance, when finding areas under curves, calculating volumes, and even solving differential equations.

Understanding symmetry can improve one’s ability to visualize mathematical concepts and approach problem-solving more effectually. It also underscores the beauty of mathematics by illustrating how certain mathematical phenomena manifest repeated patterns.

Continuous Functions

A continuous function has a graph that is unbroken or seamless, and they play a crucial role in calculus and mathematical analysis. When we consider a function to be continuous within a given interval, we are noting these details:

• The function has no interruptions, holes, jumps, or breaks within the specified interval.
• Mathematically, for all points x in an interval, the limit of f(x) as x approaches that point exists and is equal to the function's value at that point.
• Continuity can often be recognized visually in a graph: if you can draw the function without lifting your pencil, then the function is continuous over the interval you’ve drawn.

Continuous functions are essential when discussing differentiability, or when solving real-world problems that require smooth transitions without sudden changes—such as modeling the motion of objects or temperature gradients. The fact that our example function f is continuous on [-3,3] adds to our understanding of its behavior and helps us confirm the functional values without unexpected variations.