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Functions - Concept of a function, domain, and range

Grade 10IB

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 (element in the domain) corresponds to exactly one output (element in the range). Visually, this is represented in a mapping diagram where only one arrow originates from each 'input' bubble, or on a graph where any vertical line drawn passes through the curve at most once (the Vertical Line Test).

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Domain refers to the set of all possible input values, typically represented by the variable xx. On a Cartesian plane, the domain is represented by the horizontal width of the graph, looking at how far the line or curve stretches from left to right along the xx-axis.

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Range refers to the set of all possible output values, typically represented by f(x)f(x) or yy. Visually, the range is shown by the vertical spread of the graph, observing the lowest and highest points reached along the yy-axis.

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Function Notation uses symbols like f(x)f(x) to represent the output value of a function ff for a specific input xx. This is not 'f times x' but rather a way to label the relationship. For example, if f(x)=2xf(x) = 2x, the point (3,6)(3, 6) on a graph indicates that f(3)=6f(3) = 6.

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Restricted Domains occur when certain values of xx make the function undefined. For example, in a rational function like f(x)=1xf(x) = \frac{1}{x}, the domain excludes x=0x = 0 because division by zero is impossible. Visually, this often results in an asymptoteβ€”a line that the graph approaches but never touches.

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Square Root Functions have domain restrictions because the value under the radical (the radicand) must be greater than or equal to zero for real numbers. For f(x)=xβˆ’af(x) = \sqrt{x-a}, the domain is xβ‰₯ax \geq a. On a graph, this appears as a curve that starts at a specific point (a,0)(a, 0) and extends in one direction.

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Discrete vs. Continuous Functions: A discrete function consists of individual, separate points (like a scatter plot), where the domain is a list of specific numbers. A continuous function is an unbroken line or curve, where the domain and range are expressed as intervals using inequalities like a≀x≀ba \leq x \leq b.

πŸ“Formulae

Function Notation: f(x)=yf(x) = y

Domain of f(x)=g(x)f(x) = \sqrt{g(x)}: Solve g(x)β‰₯0g(x) \geq 0

Domain of f(x)=1g(x)f(x) = \frac{1}{g(x)}: Solve g(x)β‰ 0g(x) \neq 0

Set Notation: {x∈R∣xβ‰ a}\{x \in \mathbb{R} \mid x \neq a\}

Interval Notation: [a,b][a, b] (includes endpoints), (a,b)(a, b) (excludes endpoints)

πŸ’‘Examples

Problem 1:

Given the function f(x)=3xβˆ’4f(x) = \frac{3}{x - 4}, determine the domain and evaluate f(7)f(7).

Solution:

  1. To find the domain, identify where the denominator is zero: xβˆ’4=0β€…β€ŠβŸΉβ€…β€Šx=4x - 4 = 0 \implies x = 4. Therefore, the domain is all real numbers except 44, written as x∈R,xβ‰ 4x \in \mathbb{R}, x \neq 4.
  2. To evaluate f(7)f(7), substitute 77 for xx: f(7)=37βˆ’4f(7) = \frac{3}{7 - 4}
  3. Simplify the expression: f(7)=33=1f(7) = \frac{3}{3} = 1

Explanation:

The domain of a rational function must exclude values that cause division by zero. Evaluation involves direct substitution of the input value into the function rule.

Problem 2:

Find the range of the function g(x)=x2+5g(x) = x^2 + 5 for the domain x∈Rx \in \mathbb{R}.

Solution:

  1. Consider the parent function y=x2y = x^2. The smallest value for x2x^2 is 00 (when x=0x = 0).
  2. Since g(x)=x2+5g(x) = x^2 + 5, the smallest value of the function occurs at the vertex. Substitute x=0x = 0: g(0)=02+5=5g(0) = 0^2 + 5 = 5.
  3. As xx increases or decreases from zero, x2x^2 grows positively, so g(x)g(x) will always be 55 or greater.
  4. The range is g(x)β‰₯5g(x) \geq 5 or [5,∞)[5, \infty).

Explanation:

For quadratic functions in the form f(x)=ax2+cf(x) = ax^2 + c where a>0a > 0, the graph is a parabola opening upwards with a minimum point at the vertex (0,c)(0, c). The range starts from this minimum yy-value and extends to infinity.