Even and Odd Functions

Even Function

An even function is a function where $f(-x) = f(x)$ for all $x$ in its domain.

• The graph is symmetric about the y-axis
• Example: $f(x) = x^2$, because $f(-x) = (-x)^2 = x^2 = f(x)$ ✓

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Odd Function

An odd function is a function where $f(-x) = -f(x)$ for all $x$ in its domain.

• The graph is symmetric about the origin
• Example: $f(x) = x^3$, because $f(-x) = (-x)^3 = -x^3 = -f(x)$ ✓

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Symmetricity Summary

Type	Condition	Symmetry
Even	$f(-x) = f(x)$	About y-axis
Odd	$f(-x) = -f(x)$	About origin

A function can also be neither even nor odd if it satisfies neither condition.

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Domain and Range

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i. $f(x) = \sqrt{5x + 10}$

Domain:

For a square root to be defined, the expression inside must be greater than or equal to zero:

$$5x + 10 \geq 0$$

$$5x \geq -10$$

$$x \geq -2$$

$$\boxed{\text{Domain} = [-2, \infty)}$$

Range:

• When $x = -2$, $f(x) = \sqrt{0} = 0$ (minimum value)
• As $x \to \infty$, $f(x) \to \infty$
• The square root always gives a non-negative output

$$\boxed{\text{Range} = [0, \infty)}$$

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ii. $f(x) = \dfrac{x^2 - 3x - 4}{x + 1}$

Domain:

The function is undefined where the denominator = 0:

$$x + 1 = 0 \Rightarrow x = -1$$

$$\boxed{\text{Domain} = \mathbb{R} \setminus \{-1\} \text{, i.e., all real numbers except } x = -1}$$

Range:

Simplify by factoring the numerator:

$$x^2 - 3x - 4 = (x-4)(x+1)$$

$$f(x) = \frac{(x-4)(x+1)}{x+1} = x - 4 \quad \text{, where } x \neq -1$$

• This is a linear function $f(x) = x - 4$, but with a hole at $x = -1$
• At $x = -1$: the value would be $-1 - 4 = -5$, but this point is excluded

$$\boxed{\text{Range} = \mathbb{R} \setminus \{-5\} \text{, i.e., all real numbers except } -5}$$