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Answer :
To determine whether each function is even, odd, or neither, let's review the definitions:
- Even Function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex]. Graphically, even functions are symmetrical about the y-axis.
- Odd Function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex]. Graphically, odd functions have point symmetry about the origin.
Now, let's analyze each given function:
39. [tex]\( h(x) = 4x^7 \)[/tex]
- Since [tex]\( h(-x) = 4(-x)^7 = -4x^7 = -h(x) \)[/tex], the function is odd.
40. [tex]\( g(x) = -2x^6 + x^2 \)[/tex]
- Evaluate [tex]\( g(-x) = -2(-x)^6 + (-x)^2 = -2x^6 + x^2 = g(x) \)[/tex]. Therefore, the function is even.
41. [tex]\( f(x) = x^4 + 3x^2 - 2 \)[/tex]
- Evaluate [tex]\( f(-x) = (-x)^4 + 3(-x)^2 - 2 = x^4 + 3x^2 - 2 = f(x) \)[/tex]. Therefore, the function is even.
42. [tex]\( f(x) = x^5 + 3x^3 - x \)[/tex]
- Evaluate [tex]\( f(-x) = (-x)^5 + 3(-x)^3 - (-x) = -x^5 - 3x^3 + x = -f(x) \)[/tex]. Therefore, the function is odd.
43. [tex]\( g(x) = x^2 + 5x + 1 \)[/tex]
- Evaluate [tex]\( g(-x) = (-x)^2 + 5(-x) + 1 = x^2 - 5x + 1 \)[/tex], which is not equal to [tex]\( g(x) \)[/tex] or [tex]\(-g(x)\)[/tex]. Therefore, the function is neither.
44. [tex]\( f(x) = -x^3 + 2x - 9 \)[/tex]
- Evaluate [tex]\( f(-x) = -(-x)^3 + 2(-x) - 9 = x^3 - 2x - 9 \)[/tex]. This is not equal to [tex]\( f(x) \)[/tex] or [tex]\(-f(x)\)[/tex]. Therefore, the function is neither.
45. [tex]\( f(x) = x^4 - 12x^2 \)[/tex]
- Evaluate [tex]\( f(-x) = (-x)^4 - 12(-x)^2 = x^4 - 12x^2 = f(x) \)[/tex]. Therefore, the function is even.
46. [tex]\( h(x) = x^5 + 3x^4 \)[/tex]
- Evaluate [tex]\( h(-x) = (-x)^5 + 3(-x)^4 = -x^5 + 3x^4 \)[/tex]. This is not equal to [tex]\( h(x) \)[/tex] or [tex]\(-h(x)\)[/tex]. Therefore, the function is neither.
These analyses show whether each function is even, odd, or neither based on their properties.
- Even Function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex]. Graphically, even functions are symmetrical about the y-axis.
- Odd Function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex]. Graphically, odd functions have point symmetry about the origin.
Now, let's analyze each given function:
39. [tex]\( h(x) = 4x^7 \)[/tex]
- Since [tex]\( h(-x) = 4(-x)^7 = -4x^7 = -h(x) \)[/tex], the function is odd.
40. [tex]\( g(x) = -2x^6 + x^2 \)[/tex]
- Evaluate [tex]\( g(-x) = -2(-x)^6 + (-x)^2 = -2x^6 + x^2 = g(x) \)[/tex]. Therefore, the function is even.
41. [tex]\( f(x) = x^4 + 3x^2 - 2 \)[/tex]
- Evaluate [tex]\( f(-x) = (-x)^4 + 3(-x)^2 - 2 = x^4 + 3x^2 - 2 = f(x) \)[/tex]. Therefore, the function is even.
42. [tex]\( f(x) = x^5 + 3x^3 - x \)[/tex]
- Evaluate [tex]\( f(-x) = (-x)^5 + 3(-x)^3 - (-x) = -x^5 - 3x^3 + x = -f(x) \)[/tex]. Therefore, the function is odd.
43. [tex]\( g(x) = x^2 + 5x + 1 \)[/tex]
- Evaluate [tex]\( g(-x) = (-x)^2 + 5(-x) + 1 = x^2 - 5x + 1 \)[/tex], which is not equal to [tex]\( g(x) \)[/tex] or [tex]\(-g(x)\)[/tex]. Therefore, the function is neither.
44. [tex]\( f(x) = -x^3 + 2x - 9 \)[/tex]
- Evaluate [tex]\( f(-x) = -(-x)^3 + 2(-x) - 9 = x^3 - 2x - 9 \)[/tex]. This is not equal to [tex]\( f(x) \)[/tex] or [tex]\(-f(x)\)[/tex]. Therefore, the function is neither.
45. [tex]\( f(x) = x^4 - 12x^2 \)[/tex]
- Evaluate [tex]\( f(-x) = (-x)^4 - 12(-x)^2 = x^4 - 12x^2 = f(x) \)[/tex]. Therefore, the function is even.
46. [tex]\( h(x) = x^5 + 3x^4 \)[/tex]
- Evaluate [tex]\( h(-x) = (-x)^5 + 3(-x)^4 = -x^5 + 3x^4 \)[/tex]. This is not equal to [tex]\( h(x) \)[/tex] or [tex]\(-h(x)\)[/tex]. Therefore, the function is neither.
These analyses show whether each function is even, odd, or neither based on their properties.
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