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Algebra - Functions: domain, range, and notation

Grade 9IB

Review the key concepts, formulae, and examples before starting your quiz.

πŸ”‘Concepts

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A function is a special type of relation where every input (xx) is paired with exactly one output (yy). Visually, in a mapping diagram, this means that every element in the starting set (domain) has exactly one arrow pointing away from it toward an element in the target set (range).

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Function notation, written as f(x)f(x), is used to name a function and show the input variable. For example, if f(x)=2xf(x) = 2x, then f(3)f(3) represents the output when 33 is the input. Visually, you can imagine a 'function machine' where xx is dropped into the top, the rule ff is applied, and the result f(x)f(x) is ejected from the bottom.

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The Domain is the set of all possible input values for which the function is defined. On a coordinate plane, the domain is the horizontal 'width' of the graph along the xx-axis. If a graph has a solid dot at an endpoint, that value is included; an open circle means the value is not included.

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The Range is the set of all possible output values that the function can produce. Visually, the range is the vertical 'height' of the graph along the yy-axis. You find it by looking for the lowest and highest points on the curve or line.

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The Vertical Line Test is a visual method used to determine if a graph represents a function. If any vertical line can be drawn such that it intersects the graph at more than one point, the relation is not a function. For example, a vertical line would hit a circle at two points, so a circle is not a function.

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Domain restrictions occur when certain values of xx would make the function undefined. This often happens in fractions where the denominator cannot be zero. Visually, these restrictions appear as gaps or 'breaks' in the graph where no line or curve exists.

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Set notation and interval symbols are used to describe domain and range precisely. For example, {x∣xβ‰₯0,x∈R}\{x | x \ge 0, x \in \mathbb{R}\} means xx is any real number greater than or equal to zero. Visually, a solid line on a graph representing all real numbers indicates a continuous domain.

πŸ“Formulae

f(x)=yf(x) = y

f:x↦ax+bf: x \mapsto ax + b

D={x∣x∈R}D = \{x | x \in \mathbb{R}\}

R={y∣y∈R}R = \{y | y \in \mathbb{R}\}

x≠0 (for f(x)=1x)x \neq 0 \text{ (for } f(x) = \frac{1}{x})

πŸ’‘Examples

Problem 1:

Given the function f(x)=5x2βˆ’3x+1f(x) = 5x^2 - 3x + 1, evaluate f(βˆ’3)f(-3).

Solution:

  1. Substitute x=βˆ’3x = -3 into the function: f(βˆ’3)=5(βˆ’3)2βˆ’3(βˆ’3)+1f(-3) = 5(-3)^2 - 3(-3) + 1 \ 2. Calculate the square of βˆ’3-3: f(βˆ’3)=5(9)βˆ’3(βˆ’3)+1f(-3) = 5(9) - 3(-3) + 1 \ 3. Perform the multiplications: f(βˆ’3)=45+9+1f(-3) = 45 + 9 + 1 \ 4. Add the terms: f(βˆ’3)=55f(-3) = 55

Explanation:

To evaluate a function for a specific value, replace the variable xx with the given number and follow the order of operations (BIDMAS/BODMAS).

Problem 2:

Determine the domain and range for the relation R={(1,2),(3,4),(5,2),(7,8)}R = \{(1, 2), (3, 4), (5, 2), (7, 8)\}. Is this relation a function?

Solution:

  1. List the first elements for the Domain: D={1,3,5,7}D = \{1, 3, 5, 7\} \ 2. List the unique second elements for the Range: R={2,4,8}R = \{2, 4, 8\} \ 3. Check for uniqueness: Each input xx (1,3,5,71, 3, 5, 7) appears only once and is assigned to exactly one output.

Explanation:

The domain consists of all unique xx-coordinates, and the range consists of all unique yy-coordinates. Because no input value is repeated with a different output, the relation is a function.