composite and inverse functions mrs. aldous, mr. beetz & mr. thauvette ib dp sl mathematics
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COMPOSITE AND INVERSE FUNCTIONSMrs. Aldous, Mr. Beetz & Mr. ThauvetteIB DP SL Mathematics
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You should be able to…
Find the composition of two functions Find the value of the composition of two
functions for a particular value of x Use the graph of a function to decide
whether the function has an inverse Sketch the graph of the inverse of a
function from the graph of a function Verify that two functions are inverses of
each other
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You should know…
The composite function can also be denoted by or more simply,
The composition is the operation of applying then ; the composition is the operation of applying then
If is a function that maps to , then the inverse function maps back onto
Geometrically, the graph of the inverse function is a reflection of the original function in the line
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You should know…
The domain of is the range of and the range of is the domain of
To find the inverse of a function analytically, simply switch the positions of x and y and then solve the resulting equation for y
A function is the inverse of if and only if
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Composite functionsA composite function is made up of two or more functions.
fg(x) means take g(x) and put it into f(x).Replace each x in f(x) with the complete g(x).
gf(x) means take f(x) and put it into g(x).Replace each x in g(x) with the complete f(x).
Try some of these:
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Inverse functionsThe inverse of a function, is reversing the operations of that function.
Replace f(x) with ‘y=’
Now make x the subject of the equation.
Finally replace y with x, and the x with a f-1(x).
Another example:
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Inverse of quadratic equationsFirst, ensure that the equation is in the completed the square form. Try some of these:
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Example
Consider the functions and
(a) Find
(b) Find
(c) Write down the domain of
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Example continued…
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Example continued…
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Example continued…
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Be prepared…
The order in which the functions are to be composed is extremely important
Proceed with caution when simplifying expressions resulting from a composition