Determine algebraically whether the function is even, odd, or neither. 11) f(x) = 8x^3 + 3 12) f(x) = -2x^2 + 4 13) f(x) = 4x^3
The Correct Answer and Explanation is:
To determine whether a function is even, odd, or neither, we use the following definitions:
- Even function: A function is even if for all xxx, f(−x)=f(x)f(-x) = f(x)f(−x)=f(x).
- Odd function: A function is odd if for all xxx, f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).
- Neither: If the function doesn’t satisfy the conditions for being even or odd, then it is neither.
11) f(x)=8×3+3f(x) = 8x^3 + 3f(x)=8×3+3
To check whether this function is even or odd, we need to compute f(−x)f(-x)f(−x) and compare it to f(x)f(x)f(x) and −f(x)-f(x)−f(x).
- First, calculate f(−x)f(-x)f(−x): f(−x)=8(−x)3+3=−8×3+3f(-x) = 8(-x)^3 + 3 = -8x^3 + 3f(−x)=8(−x)3+3=−8×3+3
- Now, compare f(−x)f(-x)f(−x) to f(x)f(x)f(x) and −f(x)-f(x)−f(x):
- f(x)=8×3+3f(x) = 8x^3 + 3f(x)=8×3+3
- −f(x)=−(8×3+3)=−8×3−3-f(x) = -(8x^3 + 3) = -8x^3 – 3−f(x)=−(8×3+3)=−8×3−3
Since f(−x)=−8×3+3f(-x) = -8x^3 + 3f(−x)=−8×3+3 is not equal to f(x)f(x)f(x) nor to −f(x)-f(x)−f(x), the function is neither even nor odd.
12) f(x)=−2×2+4f(x) = -2x^2 + 4f(x)=−2×2+4
Again, we compute f(−x)f(-x)f(−x) and compare it to f(x)f(x)f(x) and −f(x)-f(x)−f(x).
- First, calculate f(−x)f(-x)f(−x): f(−x)=−2(−x)2+4=−2×2+4f(-x) = -2(-x)^2 + 4 = -2x^2 + 4f(−x)=−2(−x)2+4=−2×2+4
- Now, compare f(−x)f(-x)f(−x) to f(x)f(x)f(x) and −f(x)-f(x)−f(x):
- f(x)=−2×2+4f(x) = -2x^2 + 4f(x)=−2×2+4
- −f(x)=−(−2×2+4)=2×2−4-f(x) = -(-2x^2 + 4) = 2x^2 – 4−f(x)=−(−2×2+4)=2×2−4
Since f(−x)=f(x)f(-x) = f(x)f(−x)=f(x), the function is even.
13) f(x)=4x3f(x) = 4x^3f(x)=4×3
Once again, we compute f(−x)f(-x)f(−x) and compare it to f(x)f(x)f(x) and −f(x)-f(x)−f(x).
- First, calculate f(−x)f(-x)f(−x): f(−x)=4(−x)3=−4x3f(-x) = 4(-x)^3 = -4x^3f(−x)=4(−x)3=−4×3
- Now, compare f(−x)f(-x)f(−x) to f(x)f(x)f(x) and −f(x)-f(x)−f(x):
- f(x)=4x3f(x) = 4x^3f(x)=4×3
- −f(x)=−(4×3)=−4×3-f(x) = -(4x^3) = -4x^3−f(x)=−(4×3)=−4×3
Since f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), the function is odd.
Summary of Results:
- f(x) = 8x^3 + 3 is neither even nor odd.
- f(x) = -2x^2 + 4 is even.
- f(x) = 4x^3 is odd.
