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VCE Math Methods Unit 2 AOS 1
Relations, Functions, Domain & Range
This section covers the essential language of functions. A relation is any set of ordered pairs, while a function is a special relation where each input has only one output. This is checked using the Vertical Line Test.
Domain
The set of all possible input values (x-values) for which a function is defined.
Range
The set of all possible output values (y-values) that a function can produce.
A Unified Theory of Transformations
All transformations can be understood with the framework . Use the controls below to see how each parameter dilates, reflects, and translates a parent function.
Equation:
Inverse Functions
An inverse function, f⁻¹(x), reverses the action of f(x). It only exists if f(x) is one-to-one. Graphically, the inverse is a reflection in the line y = x.
Click the buttons below to explore the properties of the inverse function.
Function:
,
Inverse:
,
Exponential Functions
Functions of the form y = A·bˣ + k model growth and decay. They are characterized by a horizontal asymptote at y = k.
Key Features of
- Parent Function: y = eˣ
- Transformations: Dilation by a factor of 2 from the x-axis, translated 1 unit left and 3 units down.
- Horizontal Asymptote: The line . The range is (-3, ∞).
- Y-Intercept: Found by setting x=0. The y-intercept is at .
Logarithmic Functions
As the inverse of exponentials, logarithmic functions like y = a·logₑ(x-h) + k are characterized by a vertical asymptote at x = h.
Key Features of
- Parent Function: y = ln(x)
- Transformations: Dilation by a factor of 2 from the x-axis, translated 2 units left and 1 unit up.
- Vertical Asymptote: The line . The domain is (-2, ∞).
- X-Intercept: Found by setting y=0. The x-intercept is at .
Circular Functions
Sine and Cosine functions model periodic behavior. Their key features are Amplitude, Period, Phase Shift, and Vertical Shift (Mean Position).
Solving Trigonometric Equations
Solving trig equations requires a systematic process to find all solutions within a specified domain. Click through the steps for solving over .
