C3 Flashcards
When is a mapping a function?
Give 2 examples.
A mapping is only a function if there is only 1 possible image.
One-to-one
Many-to-one
What is the:
Domain
Range
Domain = all possible input values. X values
Range = all possible output values from the input values. Y values
What do the following transformations do:
- y = f(x) + 4
- y = f(x+4)
- y = f(4x)
- y = 4f(x)
- y = -f(x)
- y = f(-x)
- Translation (0,4)
- Translation (-4,0)
- Stretch parallel to the x axis. S.F 1/4
- Stretch parallel to the y axis. S.F 4
- Reflection in x axis
- Reflection in y axis
When is a function:
- Even
- Odd
- Periodic
- Function is even if y axis is line of symmetry
f(x) = f(-x) - Function is odd if the graph has rotational symmetry of order 2 about the origin
f(-x) = -f(x) - A function is periodic if it has a repeating pattern.
Graphically, what does f^-1(x) look like?
f^-1 (x) is a reflection of f(x) in y=x
How do you differentiate:
- y = ae^x
- y = e^ax
- dy/dx = ae^x
2. dy/dx = ae^ax
How do you differentiate:
- y = alnx
- y = ln(ax)
- dy/dx = a/x
2. dy/dx = 1/x
Describe the graph of y = logx
Goes through (1,0) Never greater than y = 1 Never equal to less than x = 0
Describe the graph y = a^x
Through (0,1)
Never reaches y = 0
Quickly steepens
What does the phrase “increasing at a rate of…” mean?
It means “over time”, so means differentiate wrt time.
Why do we differentiate inverse functions?
Sometimes it is possible to differentiate the inverse function, but not the original function. You can use this derivative to find the derivative of the original.
In terms of inverse functions, what is the gradient of f(x) at x=a?
1 / gradient of f^-1(x) at x=a
What are the differentiated trig functions for:
- sinx
- cosx
- tanx
- d/dx sinx = cosx
- d/dx cosx = -sinx
- d/dx tanx = d/dx (sinx/cosx) = sec^2x
You will find this in the formula book!
What are the reciprocal trig functions for:
- 1/cosx
- 1/sinx
- 1/tanx
- 1/cosx = secx
- 1/sinx = cosecx
- 1/tanx = cotx
1st letter in the 1st word matches the 3rd letter in the 2nd word.
What is an implicit function?
A function specified by an equation containing x and y where y is not the subject
