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

3) f'(x) = x + 10

4) f'(x) = 6x - 18

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

6) f'(x) = ⅛x or
`\displaystyle\mathsf{f'(x)=(x)/(8)}`

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)=\:(x)/(6)\:+\:3}`

Replace f(x) with y:

`\displaystyle\mathsf{y=\:(x)/(6)\:+\:3}`

Switch x and y:

`\displaystyle\mathsf{x=\:(y)/(6)\:+\:3}`

Subtract 3 from both sides:

`\displaystyle\mathsf{x-3=\:(y)/(6)\:+\:3-3}`

`\displaystyle\mathsf{x-3=\:(y)/(6)}`

Multiply both sides by 6 to isolate y:

`\displaystyle\mathsf{6(x-3)=\:(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{(1)/(3)x-(7)/(3)=y}`

Replace y with f'(x):

`\displaystyle\mathsf{f'(x)=(1)/(3)x-(7)/(3)}` or
`\displaystyle\mathsf{f'(x)=(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{(x)/(8)=y}`

Replace y with f'(x):

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