Question

For questions 3 - 6, find F'(x), the inverse of F(x).

3. F(x) = x-10

4. F (x) = x/6 + 3

5. F(x) = 3x + 7

6. F(x) = 8x

Answer 1

Answer:

3)  f'(x) = x + 10

4)  f'(x) = 6x - 18

5)  $\displaystyle\mathsf{f'(x)=\frac{1}{3}x-\frac{7}{3}}$ or  $\displaystyle\mathsf{f'(x)=\frac{x - 7}{3}}$

6)  f'(x) = ⅛x  or  $\displaystyle\mathsf{f'(x)=\frac{x}{8}}$  

Step-by-step explanation:

3. f(x) = x - 10

Step 1: In order to find the inverse function of f(x) = x - 10, start by replacing f(x) with y.  

y = x - 10

Step 2:  Switch x and y:

x = y - 10

Step 3: Add 10 to both sides to isolate y:

x + 10 = y - 10 + 10

x + 10 = y

Step 4: Replace y with f'(x):

f'(x) = x + 10  ⇒ This is the inverse function of f(x).

4.  $\displaystyle\mathsf{f(x)=\:\frac{x}{6}\:+\:3}$

Replace f(x) with y:

$\displaystyle\mathsf{y=\:\frac{x}{6}\:+\:3}$

Switch x and y:

$\displaystyle\mathsf{x=\:\frac{y}{6}\:+\:3}$

Subtract 3 from both sides:

$\displaystyle\mathsf{x-3=\:\frac{y}{6}\:+\:3-3}$

$\displaystyle\mathsf{x-3=\:\frac{y}{6}}$

Multiply both sides by 6 to isolate y:

$\displaystyle\mathsf{6(x-3)=\:\frac{y}{6}(6)}$

6x - 18 = y

Replace y with f'(x):

f'(x) = 6x - 18  ⇒ This is the inverse function of f(x).

5. f(x) = 3x + 7

Replace f(x) with y:

y = 3x + 7

Switch x and y:

x = 3y + 7

Subtract 7 from both sides:

x - 7 = 3y + 7 - 7

x - 7 = 3y

Multiply both sides by ⅓:

⅓(x - 7) = 3y (⅓)

$\displaystyle\mathsf{\frac{1}{3}x-\frac{7}{3}=y}$

Replace y with f'(x):

$\displaystyle\mathsf{f'(x)=\frac{1}{3}x-\frac{7}{3}}$  or  $\displaystyle\mathsf{f'(x)=\frac{x - 7}{3}}$   ⇒ This is the inverse function of f(x).

6. f(x) = 8x

Replace f(x) with y:

y = 8x

Switch x and y:

x = 8y

Multiply both sides by ⅛:

⅛(x) = ⅛(8y)

⅛x = y or  $\displaystyle\mathsf{\frac{x}{8}=y}$  

Replace y with f'(x):

f'(x) = ⅛x  or $\displaystyle\mathsf{f'(x)=\frac{x}{8}}$   ⇒ This is the inverse function of f(x).