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Algebra - Functions, Mapping Notation, Domain, and Range

Grade 8IB

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

🔑Concepts

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A Function is a special relationship where each input (from the domain) is paired with exactly one output (in the range). If an input maps to multiple outputs, it is not a function.

Mapping diagram showing inputs a and b mapping to unique outputs f(a) and f(b).
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Mapping Notation: We use f:x↦ax+bf: x \mapsto ax + b to denote that the function ff maps the input xx to the expression ax+bax + b. This is equivalent to writing f(x)=ax+bf(x) = ax + b.

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Domain: The set of all possible input values (xx-values) for which the function is defined.

Graph of y = x^2 showing that the domain extends infinitely along the x-axis.
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Range: The set of all possible output values (yy-values) resulting from the domain values.

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Vertical Line Test: A curve in the coordinate plane is a function of xx if and only if no vertical line intersects the curve more than once.

📐Formulae

f(x)=yf(x) = y

f:x↦ax+bf: x \mapsto ax + b

Output=Function(Input)\text{Output} = \text{Function}(\text{Input})

Domain→Function→Range\text{Domain} \rightarrow \text{Function} \rightarrow \text{Range}

💡Examples

Problem 1:

Given the function f(x)=3x2−5f(x) = 3x^2 - 5, find the value of f(4)f(4).

Solution:

f(4)=3(4)2−5f(4) = 3(4)^2 - 5 f(4)=3(16)−5f(4) = 3(16) - 5 f(4)=48−5f(4) = 48 - 5 f(4)=43f(4) = 43

Explanation:

Substitute the value x=4x = 4 into the function expression and follow the order of operations (BODMAS/BIDMAS).

Problem 2:

A function is defined by g:x↦x+102g: x \mapsto \frac{x + 10}{2}. Find the range for the domain {−2,0,4}\{ -2, 0, 4 \}.

Solution:

For x=−2x = -2: g(−2)=−2+102=82=4g(-2) = \frac{-2 + 10}{2} = \frac{8}{2} = 4 For x=0x = 0: g(0)=0+102=102=5g(0) = \frac{0 + 10}{2} = \frac{10}{2} = 5 For x=4x = 4: g(4)=4+102=142=7g(4) = \frac{4 + 10}{2} = \frac{14}{2} = 7 Range={4,5,7}\text{Range} = \{ 4, 5, 7 \}

Explanation:

To find the range, substitute each element of the domain into the mapping rule to find the corresponding outputs.

Problem 3:

Identify the domain and range from the set of ordered pairs: S={(1,5),(2,10),(3,15),(4,20)}S = \{ (1, 5), (2, 10), (3, 15), (4, 20) \}.

Solution:

Domain={1,2,3,4}\text{Domain} = \{ 1, 2, 3, 4 \} Range={5,10,15,20}\text{Range} = \{ 5, 10, 15, 20 \}

Explanation:

In a set of ordered pairs (x,y)(x, y), the domain consists of all the first coordinates (xx), and the range consists of all the second coordinates (yy).

Problem 4:

Given the function f(x)=x2−2f(x) = x^2 - 2 with domain D={−2,−1,0,1,2}D = \{ -2, -1, 0, 1, 2 \}, determine the range and sketch the mapping.

Mapping diagram from Domain {-2, -1, 0, 1, 2} to Range {-2, -1, 2}.

Solution:

f(−2)=(−2)2−2=4−2=2f(-2) = (-2)^2 - 2 = 4 - 2 = 2 f(−1)=(−1)2−2=1−2=−1f(-1) = (-1)^2 - 2 = 1 - 2 = -1 f(0)=(0)2−2=−2f(0) = (0)^2 - 2 = -2 f(1)=(1)2−2=−1f(1) = (1)^2 - 2 = -1 f(2)=(2)2−2=2f(2) = (2)^2 - 2 = 2 Range: R={−2,−1,2}R = \{ -2, -1, 2 \}

Explanation:

To find the range, substitute each element of the domain into the function. Note that even though the domain has 5 elements, the range only has 3 because some outputs are repeated.

Problem 5:

Determine the domain and range of the linear function h(x)=2x+1h(x) = 2x + 1 shown on the coordinate plane for the interval 0≤x≤40 \le x \le 4.

Line segment from (0,1) to (4,9) representing the function h(x) = 2x + 1.

Solution:

From the graph: Minimum x=0x = 0, Maximum x=4x = 4. Domain: 0≤x≤40 \le x \le 4. Minimum y=h(0)=2(0)+1=1y = h(0) = 2(0) + 1 = 1. Maximum y=h(4)=2(4)+1=9y = h(4) = 2(4) + 1 = 9. Range: 1≤y≤91 \le y \le 9.

Explanation:

For a continuous function on a closed interval, the range is determined by finding the outputs of the minimum and maximum domain values.

Functions, Mapping Notation, Domain, and Range Grade 8 Notes & Examples