Determine if Odd, Even, or Neither f(x)=4x^2-4x+4

Problem

ƒ(x)=4*x2−4*x+4

Solution

1. Recall the definitions for function parity: a function is even if ƒ*(−x)=ƒ(x) odd if ƒ*(−x)=−ƒ(x) and neither if neither condition is met.

2. Substitute −x for x in the original function to evaluate ƒ*(−x)

ƒ*(−x)=4*(−x)2−4*(−x)+4

3. Simplify the expression by squaring the negative variable and performing the multiplication.

ƒ*(−x)=4*x2+4*x+4

4. Compare ƒ*(−x) to the original function ƒ(x)

4*x2+4*x+4≠4*x2−4*x+4

⇒ƒ*(−x)≠ƒ(x)

5. Compare ƒ*(−x) to −ƒ(x) by factoring out a negative sign from the original function.

−ƒ(x)=−(4*x2−4*x+4)=−4*x2+4*x−4

4*x2+4*x+4≠−4*x2+4*x−4

⇒ƒ*(−x)≠−ƒ(x)

6. Conclude that since ƒ*(−x) is equal to neither ƒ(x) nor −ƒ(x) the function is neither even nor odd.

Final Answer

ƒ(x)=4*x2−4*x+4* is Neither

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