Functions

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Everything Maths www.everythingmaths.co.za
6. Functions
Grade 10

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Functions
● A function is a mathematical relationship between two variables, where every input variable
has one output variable.
● The x-variable is known as the input or independent variable, because its value can be
chosen freely.
● The calculated y-variable is known as the output or dependent variable, because its value
depends on the chosen input value
● The domain of a function is the set of all x-values for which there exists at most one y-value
according to that function.
● The range is the set of all y-values which can be obtained using at least one x-value.
● An asymptote is a straight line, which the graph of a function will approach, but never touch.
● A graph is said to be continuous if there are no breaks in the graph.

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Different notation
● Set notation e.g:
● Interval notation e.g:
● Function notation e.g:
f (x ) or g (x ) or h (x )

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Representing functions
● Words: “The relationship between two variables is such that one is always 5 less than the
other”
● Mapping diagram:
● Table:

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Representing functions contd...
● Set of ordered number pairs: (-3; -8), (0;-5), (5;0)
● Algebraic formula: f(x) = x - 5
● Graph:

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Linear functions
● Basic equation: y = mx + c
● m affects the slope of the graph.
● c affects where the graph cuts the y-axis
● Domain:
● Range:

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Quadratic functions
● Basic equation: y = ax2
+ q
● q shifts the graph vertically upwards or downwards
● a determines the shape of the graph
● Domain:
● Range: a > 0
a < 0
● Turning points: a > 0 graph is a “smile” and has a
minimum turning point at (0;q)
a < 0 graph is a “smile” and has a
maximum turning point at (-q;0)
● Axis of symmetry: y-axis

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Hyperbolic functions
● Basic equation:
● Horizontal asymptote: y = q
● Vertical asymptote: y-axis
● Domain:
● Range:
● Axes of symmetry: y = x + q and y = -x + q
y=
a
x
+q

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Exponential functions
● Basic equation: y = abx
+ q
● Domain:
● Range: a < 0
a > 0
● Asymptotes: x = q

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Sine functions
● Basic equation: y = a sin θ + q
● Domain: [0°; 360°]
● Range: a > 0
a < 0
● Period: 360°

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Cosine functions
● Basic equation: y = a cos θ + q
● Domain: [0°; 360°]
● Range: a > 0
a < 0
● Period: 360°

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Tangent functions
● Basic equation: y = a tan θ + q
● Domain:
● Range:
● Period: 180°
● Asymptotes: θ = 90° and θ = 270°

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Functions

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Functions