Core 3 - Functions Flashcards
What are the two types of functions? Give appropriate examples. What do these functions produce?
Many - 1 e.g. y=sin(x), y=x^2. Many inputs give one y output
1-1 e.g. y=2x+3 One input gives one y output.
What type of relationship cannot be considered a function? Give an example. What do these relationships produce?
1-many relationships produce more than one output for each x input.e.g. x=y^2
What is the domain of a function?
The x values for which the function is defined.
What is the range of a function?
The y values for the suitable x inputs.
How do you find the inverse function of f(x)=X^2 +3
Switch x and y around: x=y^2 +3
Solve for y: y=sqrt(x-3)
Replace y with f^-1(x).
For a composite function fg(x), what is the order of substitution? Give the equation of fg(x)
f(x)=3x+4
g(x)=x^2
Substitute g(x) into f(x)
fg(x)=3x^2 +4
For composite functions, how do we work out the domain and range?
Find the most limiting domain of the functions used, as this will be the domain of the composite function. Substitute the extreme values into the equation to find the range.
When finding the inverse function, what is the relationship between domain and range, and what is the final step after having found the inverse function in terms of y?
The domain and range are swapped, as the inverse function is a reflection in y=x. Domain becomes range, and vice versa.
Remember to change the notation from f(x) to f^-1(x) to indicate the inverse function.
When stating the domain and range of a function, which notation do we use in the inequality?
Domain
use x in the inequality
Range Use f(x) in the inequality to indicate an output
What are the rules for translations of functions?
Think what X or Y is being replaced by. Then do the translations one step at a time, using opposite of bidmas.
If there is a squared term, when finding an inverse, what must there be?
2 solutions. Remember, if you square root, it’s the +-root
When solving inverse functions involving squared terms, what must you do?
Check each solution satisfies the domain of the equation, and explicitly exclude values that don’t match.
What is the sequence of transformations that maps y=f(x) onto y=1/3 f(x)+1?
replace y with 3y
Replace y with y-1 hence 3(y-1)
so…
Stretch factor 1/3 in y direction
translation by [0,1]
What is the sequence of transformations that maps y=g(x) onto y=-f(x)+5
y replaced by -y
replace y with y-5 hence -(y-5)
What is the sequence of transformations that maps y=h(x) into y=h(2x-8)
replace x with x-8
replace x with 2x
hence translation by [+8,0]
stretch factor 1/2 in x direction
Or…
replace x with 2x
replace x with x-4
hence 2(x-4)=2x-8