Functions

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.

img_2629

Graphically:

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?

img_2630

2. Composite functions.

img_4928

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)

= 2(x-1)

= 2x-2

(ii) g[f(x)]= g(2x)

=2x-1

Note: Generally fg≠gf (not commutative)

 

3. Inverse functions

img_4929

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.

Let y=f(x)

img_2635

Note:

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.

img_2636

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.

Answer:

(i) f(x)= 2x-3

ƒ-1 (x)= (x+3)/2

(ii) img_2638

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.

 

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