Question

Is the cosine function even, odd, or neither? Is its graph symmetric? With respect to what?

Short answer

The cosine function is even and its graph is symmetric with respect to the y-axis.

Step-by-step solution

- Define Even and Odd Functions

A function is even if for every x in its domain, \( f(-x) = f(x) \). A function is odd if for every x in its domain, \( f(-x) = -f(x) \).

- Apply the Even Function Test

Consider the cosine function \( f(x) = \cos(x) \). Substitute \(-x\) into the function and see if \( f(-x) = f(x) \): \cos(-x) = \cos(x) \.

- Verify the Symmetry

Since \( \cos(-x) = \cos(x) \) for all x, the cosine function is even. Therefore, its graph is symmetric with respect to the y-axis. Symmetry with respect to the y-axis means that \( \cos(x) = \cos(-x) \) holds true.

Key concepts

Even Function

Understanding the concept of an even function is crucial for examining the symmetry properties of various mathematical functions. In general, a function is termed 'even' if it satisfies the condition:

\[ f(-x) = f(x) \]

for all values of x in its domain. This means that the function produces the same output for both positive and negative inputs. A classic example is the cosine function, where \( f(x) = \cos(x) \). By substituting \(-x\) into the cosine function, we get

\[ \cos(-x) = \cos(x) \]

which holds true for all x. Hence, the cosine function is an even function.

Function Symmetry

Function symmetry is a significant concept when analyzing the graphs of mathematical functions. Symmetry can help us predict and understand the behavior of functions without plotting every single point. An even function, which we discussed earlier, is symmetric with respect to the y-axis. This means that the left side of the graph mirrors the right side.

If we have a function \( f(x) \) and replace x with \(-x\), the function maintains its values if it is symmetric, i.e.,

\[ f(-x) = f(x) \]

By applying this property to the cosine function, we find:

\[ \cos(-x) = \cos(x) \]

This implies that the graph of the cosine function is symmetric with respect to the y-axis. Recognizing this pattern can simplify many problems and make it easier to visualize the function's behavior.

Y-Axis Symmetry

Y-axis symmetry, also known as reflection symmetry, occurs when a function's graph remains unchanged after reflecting it over the y-axis. In simpler terms, if you were to fold the graph along the y-axis, both halves would overlap perfectly.

This type of symmetry is a characteristic of even functions. To determine if a function has y-axis symmetry, we check if it meets the condition:

\[ f(-x) = f(x) \]

The cosine function is a perfect example of a function with y-axis symmetry. For every angle \( x \), the value of the cosine is the same for \( -x \) as it is for \( x \), proving

\[ \cos(-x) = \cos(x) \]

The graph of \( \cos(x) \) clearly exhibits this symmetry by showing identical bell-shaped waves extending on either side of the y-axis.