How do you determine if f(x) = 7 is an even or odd function?

To determine if the function \( f(x) = 7 \) is even, odd, or neither, we can use the definitions of even and odd functions.

Definition of Even and Odd Functions

• Even Function: A function is even if \( f(-x) = f(x) \) for all \( x \) in its domain.
• Odd Function: A function is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.

Testing the Function

Let's apply these definitions to \( f(x) = 7 \).

Step 1: Check for Evenness

Calculate \( f(-x) \):

Since \( f(x) = 7 \), we have:

\( f(-x) = 7 \)

Now, compare \( f(-x) \) and \( f(x) \):

\( f(-x) = 7 \) and \( f(x) = 7 \)

Since \( f(-x) = f(x) \), the function is even.

Step 2: Check for Oddness

Now, calculate \( -f(x) \):

\( -f(x) = -7 \)

Compare \( f(-x) \) and \( -f(x) \):

\( f(-x) = 7 \) and \( -f(x) = -7 \)

Since \( f(-x) \neq -f(x) \), the function is not odd.

Final Assessment

Based on the tests, \( f(x) = 7 \) is an even function because it satisfies the condition \( f(-x) = f(x) \). It is not odd, as it does not meet the criteria for odd functions.