Worksheet on identifying even and odd functions with solutions.

To determine whether a function is even, odd, or neither, we use the following definitions:

- A function $ f(x) $ is even if $ f(-x) = f(x) $ for all $ x $ in the domain.
- A function $ f(x) $ is odd if $ f(-x) = -f(x) $ for all $ x $ in the domain.
- If neither condition holds, the function is neither even nor odd.

We will evaluate each function:

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k) $ f(x) = x^2 + x - 7 $

Compute $ f(-x) $:
$$
f(-x) = (-x)^2 + (-x) - 7 = x^2 - x - 7
$$
Compare with $ f(x) $ and $ -f(x) $:
- $ f(-x) = x^2 - x - 7 \ne f(x) = x^2 + x - 7 $
- $ -f(x) = -x^2 - x + 7 \ne f(-x) $

So, neither even nor odd.

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l) $ f(x) = x^4 + 5x^2 $

Compute $ f(-x) $:
$$
f(-x) = (-x)^4 + 5(-x)^2 = x^4 + 5x^2 = f(x)
$$
So, $ f(-x) = f(x) $ → even.

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m) $ f(x) = x^3 + x^2 $

Compute $ f(-x) $:
$$
f(-x) = (-x)^3 + (-x)^2 = -x^3 + x^2
$$
Compare:
- $ f(-x) = -x^3 + x^2 \ne f(x) = x^3 + x^2 $
- $ -f(x) = -x^3 - x^2 \ne f(-x) $

So, neither even nor odd.

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n) $ f(x) = |x| + x $

Compute $ f(-x) $:
$$
f(-x) = |-x| + (-x) = |x| - x
$$
Compare:
- $ f(-x) = |x| - x \ne f(x) = |x| + x $
- $ -f(x) = -|x| - x \ne f(-x) $

So, neither even nor odd.

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o) $ f(x) = \frac{x}{x^2 + 2} $

Compute $ f(-x) $:
$$
f(-x) = \frac{-x}{(-x)^2 + 2} = \frac{-x}{x^2 + 2} = -\frac{x}{x^2 + 2} = -f(x)
$$
So, $ f(-x) = -f(x) $ → odd.

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p) $ f(x) = \frac{x^4 + 3}{x^2 + 1} $

Compute $ f(-x) $:
$$
f(-x) = \frac{(-x)^4 + 3}{(-x)^2 + 1} = \frac{x^4 + 3}{x^2 + 1} = f(x)
$$
So, $ f(-x) = f(x) $ → even.

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q) $ f(x) = \frac{1}{x^3 - 2} $

Compute $ f(-x) $:
$$
f(-x) = \frac{1}{(-x)^3 - 2} = \frac{1}{-x^3 - 2}
$$
Compare:
- $ f(-x) = \frac{1}{-x^3 - 2} \ne f(x) = \frac{1}{x^3 - 2} $
- $ -f(x) = -\frac{1}{x^3 - 2} = \frac{-1}{x^3 - 2} $

Note that $ f(-x) = \frac{1}{-x^3 - 2} = \frac{1}{-(x^3 + 2)} = -\frac{1}{x^3 + 2} $, while $ -f(x) = -\frac{1}{x^3 - 2} $. These are not equal.

So, neither even nor odd.

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r) $ f(x) = x^2 \sqrt{x^2 + 1} $

Compute $ f(-x) $:
$$
f(-x) = (-x)^2 \sqrt{(-x)^2 + 1} = x^2 \sqrt{x^2 + 1} = f(x)
$$
So, $ f(-x) = f(x) $ → even.

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Final Answers:

- k) Neither
- l) Even
- m) Neither
- n) Neither
- o) Odd
- p) Even
- q) Neither
- r) Even