I need help determining whether the following functions are even, odd, or neither. Please help me.
1. f(x)=4x+5
2. f(x)=x^3-x-2
3. f(x)=x^4-x / x^5-x
4. f(x)= x^3-x / x^5
Remember that in order for a function to be even f(x)=f(-x) and in order to be odd you must make the whole function negative to be equal to f(-x)
The first one is neither , if you substitute the x by -x is not the same function as the one you presented in the beginning and is not odd because y I multiply the whole function by -1 it is not equal to f(-x).
2. Is neither
even: f(x)=f(-x)
odd: f(x)=-f(-x)
a) 4(3)+5 ? 4(-3)+5
17? [7 neither
b) (3)^2 -3-2 ? (-3)^3 -(-3)-2
4 ? 9 neither
c) (2^4-2)/( 2^5 -2) ? ((-2)^4-2)/((-2)^5 -(-2))
14/30 ? ( - 14/30) even
do the last on these same way...
If all the powers are odd, f(x) is odd. If all even, f(x) is even.
remember that constants are x^0, or even powers.
To determine whether a function is even, odd, or neither, we need to analyze how the function behaves under reflections across the y-axis and the origin.
1. f(x) = 4x + 5:
- Reflecting across the y-axis: Replace "x" with "-x" in the function.
f(-x) = 4(-x) + 5 = -4x + 5
- Since f(-x) ≠ f(x), the function is not even.
- Reflecting across the origin: Replace "x" with "-x" and negate the entire expression.
-f(-x) = -(4(-x) + 5) = 4x - 5
- Since -f(-x) ≠ f(x), the function is not odd.
- Therefore, the function is neither even nor odd.
2. f(x) = x^3 - x - 2:
- Reflecting across the y-axis:
f(-x) = (-x)^3 - (-x) - 2 = -x^3 + x - 2
- Since f(-x) = -f(x), the function is odd.
- Reflecting across the origin:
-f(-x) = -(-x)^3 + (-x) - 2 = -(-x^3 + x - 2) = x^3 - x + 2
- Since -f(-x) ≠ f(x), the function is not even.
- Therefore, the function is odd.
3. f(x) = (x^4 - x) / (x^5 - x):
- Reflections across the y-axis or the origin do not affect the nature of the function.
- To determine whether the function is even or odd, substitute "-x" for "x" and simplify:
f(-x) = ((-x)^4 - (-x)) / ((-x)^5 - (-x)) = (x^4 + x) / (x^5 + x)
- Notice that f(-x) = f(x), meaning the function is even.
- Therefore, the function is even.
4. f(x) = (x^3 - x) / (x^5):
- Reflections across the y-axis or the origin do not affect the nature of the function.
- Let's substitute "-x" for "x" and see if the function remains the same:
f(-x) = (-x^3 - (-x)) / ((-x)^5) = (-x^3 + x) / (x^5)
- Notice that f(-x) = -f(x), meaning the function is odd.
- Therefore, the function is odd.
In summary:
1. f(x) = 4x + 5 is neither even nor odd.
2. f(x) = x^3 - x - 2 is odd.
3. f(x) = (x^4 - x) / (x^5 - x) is even.
4. f(x) = (x^3 - x) / (x^5) is odd.