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Answer :
To determine whether each function is even, odd, or neither, we can evaluate the functions using the following definitions:
1. A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
2. A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
3. If neither condition is satisfied, the function is neither.
Let's analyze each function:
39. [tex]\( h(x) = 4x^7 \)[/tex]
Check for oddness:
- [tex]\( h(-x) = 4(-x)^7 = -4x^7 \)[/tex]
- Since [tex]\( h(-x) = -h(x) \)[/tex], the function is odd.
40. [tex]\( g(x) = -2x^6 + x^2 \)[/tex]
Check for evenness:
- [tex]\( g(-x) = -2(-x)^6 + (-x)^2 = -2x^6 + x^2 \)[/tex]
- Since [tex]\( g(-x) = g(x) \)[/tex], the function is even.
41. [tex]\( f(x) = x^4 + 3x^2 - 2x \)[/tex]
Check both conditions:
- [tex]\( f(-x) = (-x)^4 + 3(-x)^2 - 2(-x) = x^4 + 3x^2 + 2x \)[/tex]
- Neither [tex]\( f(-x) = f(x) \)[/tex] nor [tex]\( f(-x) = -f(x) \)[/tex] is true, so the function is neither.
42. [tex]\( f(x) = x^5 + 3x^3 - x \)[/tex]
Check for oddness:
- [tex]\( f(-x) = (-x)^5 + 3(-x)^3 - (-x) = -x^5 - 3x^3 + x \)[/tex]
- Since [tex]\( f(-x) = -f(x) \)[/tex], the function is odd.
43. [tex]\( g(x) = x^2 + 5x + 1 \)[/tex]
Check both conditions:
- [tex]\( g(-x) = (-x)^2 + 5(-x) + 1 = x^2 - 5x + 1 \)[/tex]
- Neither [tex]\( g(-x) = g(x) \)[/tex] nor [tex]\( g(-x) = -g(x) \)[/tex] is true, so the function is neither.
44. [tex]\( f(x) = -x^3 + 2x - 9 \)[/tex]
Check both conditions:
- [tex]\( f(-x) = -(-x)^3 + 2(-x) - 9 = x^3 - 2x - 9 \)[/tex]
- Neither [tex]\( f(-x) = f(x) \)[/tex] nor [tex]\( f(-x) = -f(x) \)[/tex] is true, so the function is neither.
45. [tex]\( f(x) = x^4 - 12x^2 \)[/tex]
Check for evenness:
- [tex]\( f(-x) = (-x)^4 - 12(-x)^2 = x^4 - 12x^2 \)[/tex]
- Since [tex]\( f(-x) = f(x) \)[/tex], the function is even.
46. [tex]\( h(x) = x^5 + 3x^4 \)[/tex]
Check both conditions:
- [tex]\( h(-x) = (-x)^5 + 3(-x)^4 = -x^5 + 3x^4 \)[/tex]
- Neither [tex]\( h(-x) = h(x) \)[/tex] nor [tex]\( h(-x) = -h(x) \)[/tex] is true, so the function is neither.
These analyses result in the following classifications for each function:
- [tex]\( h(x) = 4x^7 \)[/tex] is odd.
- [tex]\( g(x) = -2x^6 + x^2 \)[/tex] is even.
- [tex]\( f(x) = x^4 + 3x^2 - 2x \)[/tex] is neither.
- [tex]\( f(x) = x^5 + 3x^3 - x \)[/tex] is odd.
- [tex]\( g(x) = x^2 + 5x + 1 \)[/tex] is neither.
- [tex]\( f(x) = -x^3 + 2x - 9 \)[/tex] is neither.
- [tex]\( f(x) = x^4 - 12x^2 \)[/tex] is even.
- [tex]\( h(x) = x^5 + 3x^4 \)[/tex] is neither.
1. A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
2. A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
3. If neither condition is satisfied, the function is neither.
Let's analyze each function:
39. [tex]\( h(x) = 4x^7 \)[/tex]
Check for oddness:
- [tex]\( h(-x) = 4(-x)^7 = -4x^7 \)[/tex]
- Since [tex]\( h(-x) = -h(x) \)[/tex], the function is odd.
40. [tex]\( g(x) = -2x^6 + x^2 \)[/tex]
Check for evenness:
- [tex]\( g(-x) = -2(-x)^6 + (-x)^2 = -2x^6 + x^2 \)[/tex]
- Since [tex]\( g(-x) = g(x) \)[/tex], the function is even.
41. [tex]\( f(x) = x^4 + 3x^2 - 2x \)[/tex]
Check both conditions:
- [tex]\( f(-x) = (-x)^4 + 3(-x)^2 - 2(-x) = x^4 + 3x^2 + 2x \)[/tex]
- Neither [tex]\( f(-x) = f(x) \)[/tex] nor [tex]\( f(-x) = -f(x) \)[/tex] is true, so the function is neither.
42. [tex]\( f(x) = x^5 + 3x^3 - x \)[/tex]
Check for oddness:
- [tex]\( f(-x) = (-x)^5 + 3(-x)^3 - (-x) = -x^5 - 3x^3 + x \)[/tex]
- Since [tex]\( f(-x) = -f(x) \)[/tex], the function is odd.
43. [tex]\( g(x) = x^2 + 5x + 1 \)[/tex]
Check both conditions:
- [tex]\( g(-x) = (-x)^2 + 5(-x) + 1 = x^2 - 5x + 1 \)[/tex]
- Neither [tex]\( g(-x) = g(x) \)[/tex] nor [tex]\( g(-x) = -g(x) \)[/tex] is true, so the function is neither.
44. [tex]\( f(x) = -x^3 + 2x - 9 \)[/tex]
Check both conditions:
- [tex]\( f(-x) = -(-x)^3 + 2(-x) - 9 = x^3 - 2x - 9 \)[/tex]
- Neither [tex]\( f(-x) = f(x) \)[/tex] nor [tex]\( f(-x) = -f(x) \)[/tex] is true, so the function is neither.
45. [tex]\( f(x) = x^4 - 12x^2 \)[/tex]
Check for evenness:
- [tex]\( f(-x) = (-x)^4 - 12(-x)^2 = x^4 - 12x^2 \)[/tex]
- Since [tex]\( f(-x) = f(x) \)[/tex], the function is even.
46. [tex]\( h(x) = x^5 + 3x^4 \)[/tex]
Check both conditions:
- [tex]\( h(-x) = (-x)^5 + 3(-x)^4 = -x^5 + 3x^4 \)[/tex]
- Neither [tex]\( h(-x) = h(x) \)[/tex] nor [tex]\( h(-x) = -h(x) \)[/tex] is true, so the function is neither.
These analyses result in the following classifications for each function:
- [tex]\( h(x) = 4x^7 \)[/tex] is odd.
- [tex]\( g(x) = -2x^6 + x^2 \)[/tex] is even.
- [tex]\( f(x) = x^4 + 3x^2 - 2x \)[/tex] is neither.
- [tex]\( f(x) = x^5 + 3x^3 - x \)[/tex] is odd.
- [tex]\( g(x) = x^2 + 5x + 1 \)[/tex] is neither.
- [tex]\( f(x) = -x^3 + 2x - 9 \)[/tex] is neither.
- [tex]\( f(x) = x^4 - 12x^2 \)[/tex] is even.
- [tex]\( h(x) = x^5 + 3x^4 \)[/tex] is neither.
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