Question

A function for which \(f(x)=f(-x)\) for all \(x\) in the domain of \(f\) is called \(\mathrm{a}(\mathrm{n})\) _____ function.

Short answer

even

Step-by-step solution

Understand the given property

The problem states that for a function, the value of the function at any point is the same as the value of the function at the corresponding negative point, i.e., the function satisfies the condition \(f(x) = f(-x)\).

Recognize the type of function

A function that satisfies the condition \(f(x) = f(-x)\) for all values of \(x\) in its domain is known as an even function. This is a fundamental property of even functions.

Review the definition of an even function

By definition, a function \(f\) is even if for every \(x\) in the domain of \(f\), \(f(x) = f(-x)\). This symmetry indicates that the graph of the function is symmetric with respect to the y-axis.

Key concepts

function symmetry

Function symmetry is a key concept in algebra and calculus. It refers to how a function behaves with respect to certain transformations like reflections or rotations. One common type is symmetry around the y-axis. Symmetric functions can help us understand the function's behavior without needing to evaluate it at every single point. This can make graphing and analyzing the function much easier. If a function is symmetric around the y-axis, it means that the graph of the function looks the same on both sides of the y-axis.
This type of symmetry simplifies our work since we only need to study half the function to understand the whole graph. For example, if you know how the function behaves for positive values of x, you automatically know how it behaves for negative values of x if it's symmetric around the y-axis.

even function definition

An even function is a specific type of function symmetry. By definition, a function is considered even if for every x in its domain, the function satisfies the property \(f(x) = f(-x)\). This means that if you take any point x and substitute it into the function, and then substitute its negative \(-x\), you will obtain the exact same result.
In mathematical terms: if \( f \) is an even function, then \( \forall x \text{ in the domain of } f, \ f(x) = f(-x) \). This is a clear and simple criterion to identify even functions. For example, consider the simple function \( f(x) = x^2 \). Testing symmetry, we get \( f(x) = x^2 \) and \( f(-x) = (-x)^2 = x^2 \). Since these are equal, \(f(x) = x^2 \) is an even function.

y-axis symmetry

Y-axis symmetry is a visual representation of an even function's property. When we say a function is symmetric with respect to the y-axis, it means that every point on the function's graph has a corresponding point mirrored across the y-axis.
This turns into an essential tool for graphing functions, as identifying symmetry can reduce the workload by half. If you can determine the function's behavior on one side of the y-axis, you can predict its behavior on the other side just by reflection. For example, if a point (a, b) lies on the graph of the function, the point (-a, b) will also lie on the graph if the function is even.
This y-axis symmetry provides a quick way to check whether a function is even. If the graph of the function looks identical on both sides of the y-axis, then the function satisfies the property \(f(x) = f(-x)\), confirming it as an even function.