Answer :
To determine whether the function [tex]\( f(x) = -3x^2 - 7x^6 - 8 \)[/tex] is even, odd, or neither, let's explore the properties of even and odd functions:
1. Even Function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
2. Odd Function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
Now, let's test [tex]\( f(x) \)[/tex]:
### Step 1: Check if the Function is Even
Calculate [tex]\( f(-x) \)[/tex] and see if it is equal to [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = -3x^2 - 7x^6 - 8 \][/tex]
Calculate:
[tex]\[ f(-x) = -3(-x)^2 - 7(-x)^6 - 8 \][/tex]
Since [tex]\( (-x)^2 = x^2 \)[/tex] and [tex]\( (-x)^6 = x^6 \)[/tex], this simplifies to:
[tex]\[ f(-x) = -3x^2 - 7x^6 - 8 \][/tex]
Since [tex]\( f(-x) = f(x) \)[/tex], the function is even.
### Step 2: Check if the Function is Odd
Calculate [tex]\( f(-x) \)[/tex] and see if it equals [tex]\(-f(x)\)[/tex]:
Using the same calculation from the even check:
[tex]\[ f(-x) = -3x^2 - 7x^6 - 8 \][/tex]
Calculate [tex]\(-f(x)\)[/tex]:
[tex]\[ -f(x) = -(-3x^2 - 7x^6 - 8) = 3x^2 + 7x^6 + 8 \][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function is not odd.
### Conclusion
Since [tex]\( f(x) \)[/tex] satisfies the condition for being an even function, the function [tex]\( f(x) = -3x^2 - 7x^6 - 8 \)[/tex] is even.
1. Even Function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
2. Odd Function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
Now, let's test [tex]\( f(x) \)[/tex]:
### Step 1: Check if the Function is Even
Calculate [tex]\( f(-x) \)[/tex] and see if it is equal to [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = -3x^2 - 7x^6 - 8 \][/tex]
Calculate:
[tex]\[ f(-x) = -3(-x)^2 - 7(-x)^6 - 8 \][/tex]
Since [tex]\( (-x)^2 = x^2 \)[/tex] and [tex]\( (-x)^6 = x^6 \)[/tex], this simplifies to:
[tex]\[ f(-x) = -3x^2 - 7x^6 - 8 \][/tex]
Since [tex]\( f(-x) = f(x) \)[/tex], the function is even.
### Step 2: Check if the Function is Odd
Calculate [tex]\( f(-x) \)[/tex] and see if it equals [tex]\(-f(x)\)[/tex]:
Using the same calculation from the even check:
[tex]\[ f(-x) = -3x^2 - 7x^6 - 8 \][/tex]
Calculate [tex]\(-f(x)\)[/tex]:
[tex]\[ -f(x) = -(-3x^2 - 7x^6 - 8) = 3x^2 + 7x^6 + 8 \][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function is not odd.
### Conclusion
Since [tex]\( f(x) \)[/tex] satisfies the condition for being an even function, the function [tex]\( f(x) = -3x^2 - 7x^6 - 8 \)[/tex] is even.
