1

Functions

îIn mathematics, a function is a relation between aîsetîof inputs and a set of permissible outputs

with the property that each input is related to exactly one output.

We define the following functions as:

f(x) = 2x+5, g(x) = x

2

-3, h(x) =(7-x)/3;

ï· In order to find out (f - h) (4), we need to find (f - h) and then evaluate the composite

function at x=4.

As f(x) = 2x+5 and h(x) =(7-x)/3;

(f- h)(x) is given by f(x) - h(x);

f(x) - h(x) = 2x+5 â(7-x)/3

= (3(2x+5) â (7 â x) ) / 3

=(6x + 15 â 7 + x ) / 3

= (7x + 8) / 3

We plug x = 4 in the above formulae to get the value of

(f - h)(4) = (7 * 4 + 8 ) / 3

= (28 + 8) / 3

= 36/3 = 12

Answer: (f - h) (4) =12.

2

ï· We have to evaluate compositions (fog)(x) and (hog)(x);

We know that (f o g) (x) =f (g(x));

So, f (g(x)) =f(x

2

-3) = 2(x

2

â 3) + 5 = 2x

2

- 6 + 5 = 2x

2

- 1;

And similarly, (hog) (x) =h (g(x)) = h(x

2

â 3) = {7- (x

2

-3)}/3= {7- x

2

+ 3}/3 = (10-x

2

)/3.

ï· No we have to transform theîg(x)îfunction so that the graph is moved 6 units to the right and

7 units down;

The function of graph moved to right is given by g(x â c) (where c is the number of units

moved) here c = 6 so we get

g(x â 6) as the function representing graph moved 6 units right

The function of graph moved down is given by g(x) â c (where c is number of units

moved down) here c = 7 so we get

g(x) - 7 as the function representing graph moved 7 units down

so our final function becomes

g(x â 6) â 7

evaluating we get

g(x â 6) â 7 = (x â 6)

2

- 3 â 7

= x

2

â 12x + 36 â 10

= x

2

â 12x + 26

3

ï· îthe inverse functionsîîf

-1

î(x) and h

-1

(x) can be computed as follows,

the inverse function exists only if the function is bijective. The input and output variable are

swapped.

f(x) = 2x+5

replacing x and f(x) and then solving for f(x) to get the inverse

x = 2*f(x) + 5

2*f(x) = x â 5

f(x) = (x â 5) / 2

so inverse is f

-1

(x) = (x â 5) / 2

h(x)=(7 - x)/3

replacing x and f(x) and then solving for f(x) to get the inverse

x={7-h(x)}/3

7 â h(x) = 3x

h(x) = 7 â 3x

so inverse is h

-1

(x) = 7 â 3x

References:

1> Elementary and Intermediate Algebra