Question

if g:R-R:g(x)=4-x and (fog)(x)=11-2x, find f(x)​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{f(x)=2x+3}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies g(x) = 4 - x \\ \\ \tt: \implies fog(x) = 11 - 2x \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies f(x) = ?$

• According to given question :

$\tt \circ \: Let \: f(x) = ax + b \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies fog(x) = f(g(x)) \\ \\ \tt: \implies 11 - 2x = f(4 - x) \\ \\ \tt: \implies 11 - 2x = a(4 - x) +b \\ \\ \tt: \implies 11 - 2x = 4a - ax + b \\ \\ \tt: \implies 11 - 2x =( 4a + b) - ax \\ \\ \text{Comparing \: both \: side} \\ \\ \tt: \implies - ax = - 2x \\ \\ \green{\tt: \implies a = 2 }\\ \\ \tt: \implies 4a + b = 11 \\ \\ \tt: \implies 4 \times 2 + b = 1 \\ \\ \green{\tt: \implies b = 3} \\ \\ \green{\tt \therefore f(x) = 2x + 3}$

Answer 2

GIVEN:

• fog( x ) = 11 - 2x
• g( x ) = 4 - x

TO FIND:

• f( x ) = ?

SOLUTION:

Let

• f( x ) = ax + b

We know that

fog( x ) = f [g( x )]

Substituting the values we have

→ 11 - 2x = f( 4 - x )

→ 11 - 2x = a(4 - x) + b

→ 11 - 2x = 4a - ax + b

→ 11 - 2x = (4a + b) - ax

By comparing

-ax = - 2x

→ a = 2

__________

4a + b = 11

Substituting the value of a

→ 8 + b = 11

→ b = 11 - 8

→ b = 3

Hence, f(x) = 2x + 3