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Determining Even or Odd Functions

Determining Even or Odd Functions

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Questions and Answers

Determine if the function f(x) = x² + 5 is Even, Odd, or Neither.

Even, since f(x) = -f(-x)
Neither, since f(x) ≠ f(-x) and f(-x) ≠ - f(x)
Odd, since f(x) = -f(-x)
Even, since f(x) = f(-x) (correct)

Determine if the function f(x) = x + 15 is Even, Odd, or Neither.

Odd, since f(x) = -f(-x)
Even, since f(x) = f(-x)
Neither, since f(x) ≠ f(-x) and f(-x) ≠ - f(x) (correct)
Odd, since f(x) = f(-x)

Determine if the function f(x) = x² + x³ + 10 is Even, Odd, or Neither.

Odd, since f(x) = -f(-x)
Neither, since f(x) ≠ f(-x) and f(-x) ≠ -f(x) (correct)
Even, since f(x) = f(-x)
Even, since f(x) = -f(-x)

Determine if the function f(x) = -x³ is Even, Odd, or Neither.

<p>Odd, since f(x) = -f(-x) (A)</p> Signup and view all the answers

Determine if the function f(x) = x² - x⁴ is Even, Odd, or Neither.

<p>Even, since f(x) = f(-x) (A)</p> Signup and view all the answers

Determine if the function f(x) = 9 + x + x² is Even, Odd, or Neither.

<p>Neither, since f(x) ≠ f(-x) and f(-x) ≠ -f(x) (D)</p> Signup and view all the answers

Determine if the function f(x) = x² + x⁶ + x¹⁰ is Even, Odd, or Neither.

<p>Even, since f(x) = f(-x) (A)</p> Signup and view all the answers

Determine if the function f(x) = x + x³ + x⁵ is Even, Odd, or Neither.

<p>Odd, since f(x) = -f(-x) (D)</p> Signup and view all the answers

Determine if the function f(x) = sin x + 2 is Even, Odd, or Neither.

<p>Neither, since f(x) ≠ f(-x) and f(-x) ≠ -f(x) (B)</p> Signup and view all the answers

Determine if the function f(x) = sin(x²) is Even, Odd, or Neither.

<p>Even, since f(x) = f(-x) (C)</p> Signup and view all the answers

Determine if the function f(x) = sin(-x) is Even, Odd, or Neither.

<p>Odd, since f(x) = -f(-x) (D)</p> Signup and view all the answers

Determine if the function f(x) = cos(x) + sin(x) * 3 is Even, Odd, or Neither.

<p>Neither, since f(x) ≠ f(-x) and f(-x) ≠ - f(x) (D)</p> Signup and view all the answers

Determine if the function f(x) = cos(x) + x is Even, Odd, or Neither.

<p>Neither, since f(x) ≠ f(-x) and f(-x) ≠ - f(x) (D)</p> Signup and view all the answers

Flashcards

Even Function

A function is even if replacing x with -x results in the original function. Mathematically, f(-x) = f(x).

Odd Function

A function is odd if replacing x with -x results in the negative of the original function. Mathematically, f(-x) = -f(x).

Neither Even nor Odd Function

A function that does not satisfy the conditions of being either even or odd.

Is f(x) = x^2 + 5 even, odd, or neither?

The function f(x) = x^2 + 5 is an even function because f(-x) = (-x)^2 + 5 = x^2 + 5 = f(x).

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Is f(x) = x + 15 even, odd, or neither?

The function f(x) = x + 15 is neither even nor odd because f(-x) = -x + 15 and -f(x) = -x - 15.

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Is f(x) = x^7 even, odd, or neither?

The function f(x) = x^7 is an odd function because f(-x) = (-x)^7 = -x^7 = -f(x).

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Is f(x) = x^3 - 15 even, odd, or neither?

The function f(x) = x^3 - 15 is an odd function because f(-x) = (-x)^3 - 15 = -x^3 - 15 = -f(x).

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Is f(x) = x^4 + x^3 + 10 even, odd, or neither?

The function f(x) = x^4 + x^3 + 10 is neither even nor odd because f(-x) = x^4 - x^3 + 10 and neither this result nor its negative matches the original function.

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Is f(x) = x^2 even, odd, or neither?

The function f(x) = x^2 is an even function because f(-x) = (-x)^2 = x^2 = f(x).

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Is f(x) = -x^2 even, odd, or neither?

The function f(x) = -x^2 is an even function because f(-x) = -(-x)^2 = -x^2 = f(x).

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Is f(x) = x^3 even, odd, or neither?

The function f(x) = x^3 is an odd function because f(-x) = (-x)^3 = -x^3 = -f(x).

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Is f(x) = -x^3 even, odd, or neither?

The function f(x) = -x^3 is an odd function because f(-x) = -(-x)^3 = x^3 = -f(x).

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Is f(x) = x^4 + x^2 + x - 5 even, odd, or neither?

The function f(x) = x^4 + x^2 + x - 5 is neither even nor odd because f(-x) = x^4 + x^2 - x - 5 and neither this result nor its negative matches the original function.

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Is f(x) = x^10 even, odd, or neither?

The function f(x) = x^10 is an even function because f(-x) = (-x)^10 = x^10 = f(x).

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Is f(x) = 16 - x - x^3 even, odd, or neither?

The function f(x) = 16 - x - x^3 is neither even nor odd because f(-x) = 16 + x + x^3 and neither this result nor its negative matches the original function.

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Is f(x) = -x^2 - x^4 even, odd, or neither?

The function f(x) = -x^2 - x^4 is an even function because f(-x) = -(-x)^2 - (-x)^4 = -x^2 - x^4 = f(x).

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Is f(x) = x^5 even, odd, or neither?

The function f(x) = x^5 is an odd function because f(-x) = (-x)^5 = -x^5 = -f(x).

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Is f(x) = -x^5 even, odd, or neither?

The function f(x) = -x^5 is also an odd function because f(-x) = -(-x)^5 = x^5 = -f(x).

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Is f(x) = 9 + x + x^2 even, odd, or neither?

The function f(x) = 9 + x + x^2 is neither even nor odd because f(-x) = 9 - x + x^2 and neither this result nor its negative matches the original function.

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Is f(x) = x^2 + x^6 + x^10 even, odd, or neither?

The function f(x) = x^2 + x^6 + x^10 is an even function because f(-x) = (-x)^2 + (-x)^6 + (-x)^10 = x^2 + x^6 + x^10 = f(x).

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Is f(x) = -x^2 - x^6 - x^10 even, odd, or neither?

The function f(x) = -x^2 - x^6 - x^10 is also an even function because f(-x) = -(-x)^2 - (-x)^6 - (-x)^10 = -x^2 - x^6 - x^10 = f(x).

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Is f(x) = sin(x) + 2 even, odd, or neither?

The function f(x) = sin(x) + 2 is neither even nor odd. It's odd because sin(-x) = -sin(x), but the constant term 2 prevents it from being a true odd function.

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Is f(x) = sin(x^2) even, odd, or neither?

The function f(x) = sin(x^2) is an even function because f(-x) = sin((-x)^2) = sin(x^2) = f(x).

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Is f(x) = sin(x^3) even, odd, or neither?

The function f(x) = sin(x^3) is an odd function because f(-x) = sin((-x)^3) = sin(-x^3) = -sin(x^3) = -f(x).

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Is f(x) = sin(-x) even, odd, or neither?

The function f(x) = sin(-x) is an odd function because f(-x) = sin(-(-x)) = sin(x) = -f(x).

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Is f(x) = sin(-x^2) even, odd, or neither?

The function f(x) = sin(-x^2) is an even function because f(-x) = sin(-(-x)^2) = sin(-x^2) = sin(x^2) = f(x).

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Is f(x) = sin(-x^3) even, odd, or neither?

The function f(x) = sin(-x^3) is an odd function because f(-x) = sin(-(-x^3)) = sin(x^3) = -f(x).

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Is f(x) = cos(x) + sin(x) - 3 even, odd, or neither?

The function f(x) = cos(x) + sin(x) - 3 is neither even nor odd. The cosine term is even, the sine term is odd, and the constant term is neither.

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Is f(x) = cos(-x) + x even, odd, or neither?

The function f(x) = cos(-x) + x is neither even nor odd. The cosine function is even, but the linear term is odd.

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Is f(x) = cos(x^2) even, odd, or neither?

The function f(x) = cos(x^2) is an even function because f(-x) = cos((-x)^2) = cos(x^2) = f(x).

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Is f(x) = cos(-x^2) even, odd, or neither?

The function f(x) = cos(-x^2) is also an even function because f(-x) = cos(-(-x)^2) = cos(-x^2) = cos(x^2) = f(x).

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Study Notes

Determining Even or Odd Functions

Even Function: A function is even if f(x) = f(-x) for all x in the domain. Graphically, an even function is symmetrical about the y-axis.
Odd Function: A function is odd if f(x) = -f(-x) for all x in the domain. Graphically, an odd function is symmetrical about the origin.
Neither: If a function does not satisfy either of the above conditions, it is neither even nor odd.

Examples of Functions

f(x) = x² + 5: Even, because f(x) = f(-x)
f(x) = x + 15: Odd, because f(x) = -f(-x)
f(x) = x⁷: Odd, because f(x) = -f(-x)
f(x) = x³ - 15: Odd, because f(x) = -f(-x)
f(x) = x² + x³ + 10: Neither
f(x) = x²: Even, because f(x) = f(-x)
f(x) = -x²: Even, because f(x) = f(-x)
f(x) = x³: Odd, because f(x) = -f(-x)
f(x) = -x³: Odd, because f(x) = -f(-x)
f(x) = x⁴ + x² + 5: Even, because f(x) = f(-x)
f(x) = x¹⁰: Even, because f(x) = f(-x)
f(x) = 16/x³: Neither
f(x) = x²- x²: Neither
f(x) = x⁵: Odd, because f(x) = -f(-x)
f(x) = -x⁵: Odd, because f(x) = -f(-x)
f(x) = 9 + x + x²: Neither
f(x) = x² + x⁶ + x¹⁰: Even, because f(x) = f(-x)
f(x) = -x² - x⁶ - x¹⁰: Odd, because f(x) = -f(-x)
f(x) = sin(x): Odd, because f(x) = -f(-x), sine is an odd function
f(x) = sin(x²): Neither
f(x) = cos(x²): Even, because cos(x²) = cos((-x)²)
f(x) = cos(-x²): Even, because cos(-x²) = cos(x²)
f(x) = cos(x) + sin(x³): Neither

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