Determine algebraically whether the given function is even, odd, or neither. -9x h(x) = 4x-2
A Odd
B Even
C Neither
The Correct Answer and Explanation is :
To determine whether the function ( h(x) = 4x – 2 ) is even, odd, or neither, we need to evaluate the function using the definitions of even and odd functions.
Definitions:
- Even Function: A function ( f(x) ) is even if for all ( x ) in the domain of the function, ( f(-x) = f(x) ).
- Odd Function: A function ( f(x) ) is odd if for all ( x ) in the domain of the function, ( f(-x) = -f(x) ).
Step 1: Check if the function is even.
For the function ( h(x) = 4x – 2 ), we need to evaluate ( h(-x) ):
[
h(-x) = 4(-x) – 2 = -4x – 2
]
Now, we compare ( h(-x) ) to ( h(x) ):
[
h(x) = 4x – 2
]
Clearly, ( h(-x) \neq h(x) ), because ( -4x – 2 \neq 4x – 2 ). This means the function is not even.
Step 2: Check if the function is odd.
Next, we check if ( h(-x) = -h(x) ). We already know from the previous step that:
[
h(-x) = -4x – 2
]
and
[
-h(x) = -(4x – 2) = -4x + 2
]
Since ( h(-x) = -4x – 2 ) and ( -h(x) = -4x + 2 ), we see that ( h(-x) \neq -h(x) ). Therefore, the function is not odd.
Conclusion:
Since the function is neither even nor odd, the correct answer is C) Neither.
300-word Explanation:
To determine whether a function is even, odd, or neither, we use algebraic tests based on how the function behaves when we replace ( x ) with ( -x ).
- Even Functions: An even function satisfies the condition that ( f(-x) = f(x) ) for all values of ( x ) in the function’s domain. In other words, the function’s graph is symmetric about the y-axis.
- Odd Functions: An odd function satisfies the condition that ( f(-x) = -f(x) ) for all ( x ) in the function’s domain. The graph of an odd function is symmetric with respect to the origin.
In our case, the given function is ( h(x) = 4x – 2 ). When we calculate ( h(-x) ), we get ( -4x – 2 ), which is not equal to ( h(x) = 4x – 2 ). Hence, the function is not even.
Furthermore, when we calculate ( -h(x) ), we find that ( -h(x) = -4x + 2 ), which is also not equal to ( h(-x) = -4x – 2 ). Hence, the function is not odd.
Since the function fails to satisfy the conditions for both even and odd functions, the correct classification for ( h(x) = 4x – 2 ) is that it is neither even nor odd. Thus, the correct answer is C) Neither.