Question

14. a) If f(x + 3)=x + 6, find f '(x).
b) If f(2x + 5)=4x + 13, find f'(x).
c) If f(3x + 4)=5x + 8, find f(x).​

Answer 1

Answer:

please write down full question

Answer 2

Step-by-step explanation:

$\green{ \boxed{ \large\sf \: SOLUTION}}$

$\large{ \pink{ \star}} \sf \large (a) \green{ \rightarrow}$

$\sf f(x + 3) \: =x + 6$

$\sf put \: \: x = x - 3 \: in \: both \: side$

$\sf \: we \: get \: \rightarrow$

$\sf \: f(x - 3 + 3) = x - 3 + 6 \:$

$\sf \: \implies \: f(x) = x - 3 \: \: \: \: \: \: \: \: \: \: \\ \sf \: \implies \: f '(x)= \frac{d}{dx} ( x - 3 ) \\ \sf \: \implies \: f '(x)= 1 - 0 = 1 \: \\ \sf \: \implies \green{\boxed{ \sf \: f '(x)= 1 }}\: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\large{ \pink\star}\sf \large (b) \green{ \rightarrow}$

$\sf f(2x+ 5) \: =4x + 13$

$\displaystyle\sf put \: \: x = \frac{x - 5}{2} \: \: in \: both \: side$

$\sf \: we \: get \: \rightarrow$

$\sf f(2 \bigg( \frac{x - 5}{2} \bigg) + 5) \: =4\bigg( \frac{x - 5}{2} \bigg)+ 13$

$\sf \: f(x - 5 + 5) = 2x - 10 + 13$

$\sf \: f(x) = 2x + 3$

$\sf \implies \: f(x) = 2x + 3 \: \: \: \: \: \: \: \: \: \: \\\sf \implies f'(x) = \frac{d}{dx} (2x +3 )$

$\sf \implies \: f'(x) = 2 + 0 = 2 \\\sf \implies \green{ \boxed{ \sf \: f'(x) = 2 }}\: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\large{ \pink{ \star}} \sf \large (c) \green{ \rightarrow}$

$\sf \: f(3x + 4)=5x + 8$

$\displaystyle \sf put \: \: x = \frac{x - 4}{3} \: in \: both \: side$

$\sf \: we \: get \: \rightarrow$

$\sf f(3 \bigg( \frac{x - 4}{3} \bigg) + 4) \: =5\bigg( \frac{x - 4}{3} \bigg)+ 8$

$\displaystyle \sf \: f(x - 4+ 4) = \frac{5}{3}(x) - \frac{20}{3} + 13$

$\displaystyle \sf \: f(x ) = \frac{5}{3}(x) - \frac{20}{3} + 13 \: \\ \\ \implies \sf \: f '(x) = \frac{d}{dx} \bigg( \frac{5}{3}(x) - \frac{20}{3} + 13 \bigg) \: \\ \\ \implies \sf \: f '(x) = \frac{5}{3} - 0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \green{ \boxed{\sf \: f '(x) = \: \frac{5}{3} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$