If [tex]$f(x) = 3|x| - 4$[/tex] for every number [tex]$x$[/tex] in the domain of [tex]$f$[/tex], then [tex]$f$[/tex] is

A. [tex]$\square$[/tex] an odd function
B. [tex]$\square$[/tex] neither an even nor an odd function
C. [tex]$\square$[/tex] both an even and an odd function
D. [tex]$\square$[/tex] an even function



Answer :

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