Functions, Relations & Coordinate Geometry Flashcards
(33 cards)
A function can only have an inverse when it is…
a one-to-one (1:1) function.
In order for a function to be 1:1, it must have…
all unique Y values
e.g no two X values/inputs result in the same Y value/output
In order for a rule to be a function, it must have…
all unique X values / no repeated X values
A relation is a rule that has…
repeated X values, i.e two points/Y values have the same X values
If a graph passes the vertical line test , it must be a…
function
If a graph passes the vertical line test, but not a the horizontal line test it must be a…
many-to-one function
If a graph passes the horizontal line test, it can be both a…
- One-to-one Function;
if it passed the vertical line test
OR - Relation with all unique Y values;
if it did not pass the vertical line test
What does the vertical line test actually test?
whether all X values are unique
(whether the graph is a function)
What does the horizontal line test actually test?
whether all Y values are unique
(whether the function is a 1:1 function)
A many-to-one function has…
multiple X values that give the same Y value
multiple inputs that result in the same output
all unique X values, but repeated Y values
A maximal or implied domain is…
the “compulsory” domain that is “built in to” the rule
i.e the values of X for which there cannot be a real Y value
e.g for y=√x (square root of x), x cannot be less than 0, as you cannot take the square root of a negative number.
Therefore, maximal/implied domain is [0, ∞)
A function is ODD when…
f(-x) = -f(x)
( f of negative x equals negative f(x)
A function is EVEN when…
f(-x) = f(x)
(f of negative x equals f of x)
Are all functions odd or even?
No. A function can be neither (odd nor even).
What is an implication of f(-x) = -f(x)?
- Reflection over the Y axis is the same as reflection over the x-axis.
- It is an odd function.
Can a function with a y int of 2 be an odd function?
No. For a function to be odd, the y int must equal 0.
How do you find the domain for the sum/difference of two functions?
e.g (f+g)(x)
By finding the overlap/intersection of f(x) and g(x)’s individual domains.
If the domains don’t overlap, then (f+g)(x) cannot exist.
For (f - g)(x), use the domain of -g(x).
How do you sketch the sum/difference of two functions?
Addition of Ordinates.
For each value of X, find both corresponding Y values, and add them together. ( X, Y1+Y2 ) is the new coordinate.
The domain of the product of two functions ( e.g (fg)(x) ) is…
the overlap of domain f(x) and domain g(x)
For a composite function, e.g (f(g (x)), to exist…
Range g(x) ⊆ Domain f(x)
( the range of g(x) must be a subset of the domain of f(x) )
(the range of the inner function must be a subset of the domain of the outer function)
What is the domain of a composite function, e.g (f (g (x )) ?
domain of (f (g (x )) = domain of g(x)
the domain of the comp function is the same as the domain of the inner function.
What is the range of a composite function, e.g (f (g (x )) ?
the range of a composite function should be found the same way you find the range for any other function.
How can you restrict g(x) so that (f (g (x )) exists?
- Find the range of g(x) such that it is now a subset of domain of f(x)
- Find the all the possible inputs/ X values that satisfy the new range.
- Express these X values as the new domain of g(x).
The range and domain of an inverse are…
opposite to the range and domain of the original function.
e.g the domain becomes the range,
and the range becomes the domain
