Answer :
Sure, let's determine whether each function is even, odd, or neither.
1. Function 1: [tex]\( f(x) = x^{-4} \)[/tex]
- This function involves a single term, [tex]\( x^{-4} \)[/tex].
- An even function satisfies the property [tex]\( f(x) = f(-x) \)[/tex]. Here, [tex]\( x^{-4} = (-x)^{-4} \)[/tex], which holds true.
- Therefore, the function [tex]\( f(x) = x^{-4} \)[/tex] is an even function. So, we enter E.
2. Function 2: [tex]\( f(x) = x^6 - 6x^8 + 3x^6 \)[/tex]
- Even function involves only even powers of [tex]\( x \)[/tex].
- The terms here are [tex]\( x^6, -6x^8 \)[/tex], and [tex]\( 3x^6 \)[/tex], which are all even powers.
- Since all the terms have even exponents, [tex]\( f(x) = f(-x) \)[/tex] for this function.
- Hence, the function [tex]\( f(x) = x^6 - 6x^8 + 3x^6 \)[/tex] is an even function. So, we enter E.
3. Function 3: [tex]\( f(x) = x^6 + 3x^8 + 2x^{-7} \)[/tex]
- This function contains a mix of different powers: [tex]\( x^6, 3x^8 \)[/tex] (even powers), and [tex]\( 2x^{-7} \)[/tex] (odd power).
- An even function has only even powers, while an odd function has only odd powers.
- Here, the function includes both even and odd powers, so it doesn't meet the criteria for being either even or odd.
- Thus, [tex]\( f(x) = x^6 + 3x^8 + 2x^{-7} \)[/tex] is neither an even nor an odd function. So, we enter N.
4. Function 4: [tex]\( f(x) = -5x^6 - 3x^8 - 2 \)[/tex]
- This function features terms like [tex]\( -5x^6, -3x^8 \)[/tex] (even powers), and a constant term [tex]\(-2\)[/tex].
- Despite the negativity of the coefficients, what matters is that all terms have even powers.
- Therefore, [tex]\( f(x) = f(-x) \)[/tex] holds, making it an even function.
- Consequently, the function [tex]\( f(x) = -5x^6 - 3x^8 - 2 \)[/tex] is an even function. So, we enter E.
So the answers for each function are:
1. E
2. E
3. N
4. E
1. Function 1: [tex]\( f(x) = x^{-4} \)[/tex]
- This function involves a single term, [tex]\( x^{-4} \)[/tex].
- An even function satisfies the property [tex]\( f(x) = f(-x) \)[/tex]. Here, [tex]\( x^{-4} = (-x)^{-4} \)[/tex], which holds true.
- Therefore, the function [tex]\( f(x) = x^{-4} \)[/tex] is an even function. So, we enter E.
2. Function 2: [tex]\( f(x) = x^6 - 6x^8 + 3x^6 \)[/tex]
- Even function involves only even powers of [tex]\( x \)[/tex].
- The terms here are [tex]\( x^6, -6x^8 \)[/tex], and [tex]\( 3x^6 \)[/tex], which are all even powers.
- Since all the terms have even exponents, [tex]\( f(x) = f(-x) \)[/tex] for this function.
- Hence, the function [tex]\( f(x) = x^6 - 6x^8 + 3x^6 \)[/tex] is an even function. So, we enter E.
3. Function 3: [tex]\( f(x) = x^6 + 3x^8 + 2x^{-7} \)[/tex]
- This function contains a mix of different powers: [tex]\( x^6, 3x^8 \)[/tex] (even powers), and [tex]\( 2x^{-7} \)[/tex] (odd power).
- An even function has only even powers, while an odd function has only odd powers.
- Here, the function includes both even and odd powers, so it doesn't meet the criteria for being either even or odd.
- Thus, [tex]\( f(x) = x^6 + 3x^8 + 2x^{-7} \)[/tex] is neither an even nor an odd function. So, we enter N.
4. Function 4: [tex]\( f(x) = -5x^6 - 3x^8 - 2 \)[/tex]
- This function features terms like [tex]\( -5x^6, -3x^8 \)[/tex] (even powers), and a constant term [tex]\(-2\)[/tex].
- Despite the negativity of the coefficients, what matters is that all terms have even powers.
- Therefore, [tex]\( f(x) = f(-x) \)[/tex] holds, making it an even function.
- Consequently, the function [tex]\( f(x) = -5x^6 - 3x^8 - 2 \)[/tex] is an even function. So, we enter E.
So the answers for each function are:
1. E
2. E
3. N
4. E
