To determine the nature of the function \( f(x) = 3|x| - 4 \), we need to check two key properties that categorize functions: evenness and oddness.
### Step 1: Check if the function is even
A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain.
Let's find \( f(-x) \) for the given function \( f(x) \):
[tex]\[
f(-x) = 3|-x| - 4
\][/tex]
Since the absolute value function \( |x| \) satisfies \( |-x| = |x| \), we have:
[tex]\[
f(-x) = 3|x| - 4
\][/tex]
Clearly, \( f(-x) = f(x) \).
So \( f(x) \) is an even function.
### Step 2: Check if the function is odd
A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.
Using the same expression for \( f(-x) \) as above:
[tex]\[
f(-x) = 3|x| - 4
\][/tex]
We need to check if this equals \( -f(x) \):
[tex]\[
-f(x) = -(3|x| - 4) = -3|x| + 4
\][/tex]
Comparing \( f(-x) = 3|x| - 4 \) and \( -f(x) = -3|x| + 4 \), we see that they are not equal.
Thus, \( f(x) \) is not an odd function.
### Conclusion
Since \( f(x) = 3|x| - 4 \) is an even function and not an odd function, the correct answer is:
D. [tex]\( \square \)[/tex] even function
