Finding a new function given a composite function (Case A : First function is given) Example 2

Posted on April 18, 2020 by user

Example 2
A function f is defined by

f : x \mapsto \frac{2}{x}

.
Find the function g if

f g : x \mapsto x^{2} + 1

.

[Note : the first function f is given]

Click here (check for example 5)

Finding a new function given a composite function (Case A : First function is given) Example 1

Posted on April 18, 2020 by user

Example 1
A function f is defined by

f : x \mapsto 2 x + 5

.
Find the function g if

f g : x \mapsto 3 x - 8

.

[Note : First function f is given]

Click here (check for example 5) to learn more about function

Composite Function (Comparison Method) Example 2

Posted on April 18, 2020 by user

Example 2
Given

f : x \mapsto 1 - x

and

g : x \mapsto p x^{2} + q

. If the composite function gf is defined by

g f : x \mapsto 3 x^{2} - 6 x + 5

, find
(a) the value of p and of q,
(b) the value of

g^{2} (- 1)

Composite Function (Comparison Method) Example 1

Posted on April 18, 2020 by user

Example 1
Given

f : x \mapsto h x + k, g : x \mapsto {(x + 1)}^{2} + 4

and

f g : x \mapsto 2 {(x + 1)}^{2} + 5.

Find
(a) the value of g²(2),
(b) the value of h and of k.

Example 3

Posted on April 18, 2020 by user

Example 3

If

f : x \mapsto 2 x + 1

and

g : x \mapsto \frac{5}{x}, x \neq 0

. Find the composite functions gf , fg and the value of gf(4).

Example 2

Posted on April 18, 2020 by user

Given $f : x \mapsto 1 - x$ and $g : x \mapsto p x^{2} + q$ . If the composite function gf is defined by $g f : x \mapsto 3 x^{2} - 6 x + 5$ , find

the value of p and of q,
the value of $g^{2} (- 1)$ .

Example 2

Posted on April 18, 2020 by user

Example 2

Functions f and g are defined by

f : x \mapsto x - 1

and

g : x \mapsto \frac{3 - x}{x + 4}

. Find
(a) the value of gf(3),
(b) the value of fg(-1 ),
(c) the composite functions fg,
(d) the composite functions gf,
(e) the composite functions g²,
(f) the composite functions f².

Example 1

Posted on April 18, 2020 by user

If f : x → x + 5 and g : x → x² +2x + 3, find

the value of gf (2),
the value of fg (2 ),
the composite functions fg,
the composite functions gf,
the composite functions g² ,
the composite functions f².

correction for part (c)

$\begin{array}{l} f g (x) = f (x^{2} + 2 x + 3) \\ = (x^{2} + 2 x + 3) + 5 \\ = x^{2} + 2 x + 3 + 5 \\ = x^{2} + 2 x + 8 \end{array}$

1.3a Composite Function

Posted on April 18, 2020 by user

Composite Function

If function

f : X \mapsto Y

,
and function

g : Y \mapsto Z

,
hence, composite function

g f : X \mapsto Z

Example:
If,

f : x \mapsto 2 x + 5

and

g : x \mapsto x^{2} - 1

, find

g f (2)

Answer:
$\begin{array}{l} f (x) = 2 x + 5 \\ f (2) = 2 (2) + 5 = 9 \end{array}$

$g f (2) = g [f (2)] = g (9)$

$\begin{array}{l} g (x) = x^{2} - 1 \\ g f (2) = g (9) = 9^{2} - 1 = 80 \end{array}$

Online Video Lessons

Youtube Video Lesson 1
Youtube Video Lesson 2
Youtube Video Lesson 3

Example 4 and 5

Posted on April 18, 2020 by user

Example 4
Given the function

f : x \mapsto 3 x + 2

, find the value of
(a)

f (2)

(b)

f (- 5)

(c)

f (\frac{1}{3})

Example 5
If

f (x) = x^{2} + 3 x + 2

, express each of the following in terms of x:
(a)

f (2 x)

(b)

f (3 x + 1)

(c)

f (x^{2})