Determine algebraically whether the function is even, odd, or neither
The Correct Answer and Explanation is:
Let’s analyze each function algebraically to determine if it’s even, odd, or neither.
1) f(x)=8×3+3f(x) = 8x^3 + 3f(x)=8×3+3
To determine if this function is even, odd, or neither, we need to evaluate f(−x)f(-x)f(−x) and compare it with f(x)f(x)f(x).
Step 1: Find 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
Step 2: Compare f(−x)f(-x)f(−x) with f(x)f(x)f(x)
- f(x)=8×3+3f(x) = 8x^3 + 3f(x)=8×3+3
- f(−x)=−8×3+3f(-x) = -8x^3 + 3f(−x)=−8×3+3
Since f(−x)≠f(x)f(-x) \neq f(x)f(−x)=f(x) and f(−x)≠−f(x)f(-x) \neq -f(x)f(−x)=−f(x), the function is neither even nor odd.
2) f(x)=−2×2+4f(x) = -2x^2 + 4f(x)=−2×2+4
Again, let’s evaluate f(−x)f(-x)f(−x).
Step 1: Find 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
Step 2: Compare f(−x)f(-x)f(−x) with f(x)f(x)f(x)
- f(x)=−2×2+4f(x) = -2x^2 + 4f(x)=−2×2+4
- f(−x)=−2×2+4f(-x) = -2x^2 + 4f(−x)=−2×2+4
Since f(−x)=f(x)f(-x) = f(x)f(−x)=f(x), this function is even.
3) f(x)=4x3f(x) = 4x^3f(x)=4×3
Let’s evaluate f(−x)f(-x)f(−x) for this function.
Step 1: Find f(−x)f(-x)f(−x)
f(−x)=4(−x)3=−4x3f(-x) = 4(-x)^3 = -4x^3f(−x)=4(−x)3=−4×3
Step 2: Compare f(−x)f(-x)f(−x) with f(x)f(x)f(x)
- f(x)=4x3f(x) = 4x^3f(x)=4×3
- f(−x)=−4x3f(-x) = -4x^3f(−x)=−4×3
Since f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), this function is odd.
Summary
- f(x)=8×3+3f(x) = 8x^3 + 3f(x)=8×3+3 is neither even nor odd.
- f(x)=−2×2+4f(x) = -2x^2 + 4f(x)=−2×2+4 is even.
- f(x)=4x3f(x) = 4x^3f(x)=4×3 is odd.
