1. Definition of a function:
A function is a special relationship where each input has a single unique output.
A One-to-One function is when every X value is mapped onto a different Y value.
A Many-to One function is when a Y value has more than 2 X values mapped to it.
To determine whether ƒ is a One to One function or Many to One function, just draw a straight vertical line. If the line intersects at exactly one point of the graph, then it is a One to One function.
Q: Which of the graphs is a graph of a function?
2. Composite functions.
If f(X)=Y and g(Y)=Z, then gf(X)=Z
(Always write the function performed last, the first).
E.g. If f(x)= 2x and g(x)=x-1, find in similar form (i) fg and (ii) gf
(i) f[g(x)]= f(x-1)
(ii) g[f(x)]= g(2x)
Note: Generally fg≠gf (not commutative)
3. Inverse functions
If the function ƒ maps x onto y, then its inverse function ƒ-
1 maps y back onto x.
E.g. The following function shows the function f(x)=3x-2. Find the inverse function.
1. A one-to-one function has an inverse but a many-to-one function does not have inverse.
2. To graphically identity if the function has an inverse, just draw a straight horizontal line. If the line intersects at more than one point, the function does not have an inverse.
3. The graphs of a function and its inverse are reflections of each other in the line y=x.
E.g. Given f(x)=2x-3
(i) Find ƒ-1
(ii) Draw the graphs of the function and its inverse on the same axis.
(i) f(x)= 2x-3
ƒ-1 (x)= (x+3)/2
Note: To draw the graph of a function and its inverse on the same axis, the scale on both axes should be the same.
4. For a function and its inverse.
Domain of f= Range of f-1
Range of f = Domian of f-1
If the curve has a turning point, it is a many-to-one function, thus the function has no inverse. Unless a domain is given for the function to make it a one-to-one function.