Quiz-1

• From the properties of inverse functions if f ^-1 (2) = 3 and f ^-1 (-3) = 6, then
  f(3) = 2 and f(6) = - 3


• Use the above to write
  f(3) = 3a + b = 2 and f(6) = 6a + b = -3


• Solve the 2 by 2 system of equations 3a + b = 2 and 6a + b = -3 to obtain
  a = - 5 / 3 and b = 7

• Since f(1) = 3, from the properties of the inverse functions, we have
  f ^-1(3) = 1


• f is odd function and therefore f(- 2) = - f(2) = -12. Hence
  f ^-1(- 12) = - 2

Let a = f ^-1(- 4). Then
f(a) = f(f ^-1(- 4)) = - 4 (Using the property f(f ^-1(x)) = x of the inverse function).
We now need to find a such that f(a) = - 4 hence the equation to solve.
ln(a) + 4 a - 8 = - 4
ln(a) = 4 - 4 a
The above equation cannot be solved analytically but its solution may be approximated graphically as the x coordinate of the point of intersection of the graphs of y = ln(x) and y = 4 - 4x as shown below.



The intersection of the two graphs is close to x = 1 which can easily be checked that it is the exact solution to the equation ln(x) = 4 - 4 x. Hence
f^-1( - 4) = 1

Explanations
First we drop the function notation and write y instead off(x). Then we solve for x and finally, swap x and y.

y= 3x-2 (subtract 2)
3x = y+2  (divide by 3)
(y+2)/ 3 = x
swap  x and y
f^-1(x) = 1/3 (x+2)

Solution: First we drop the function notation and write y instead off(x). Then we solve for x and finally, swap x and y.

y = (x+2) / (4x - 4) (multiply by 4x-4)
y(4x-4) = x+2 (distribute)
4xy – 4y = x+2 (add 4y, subtract x)
4xy – x = 4y +2 (factor out x)
x(4y – 1) = 4y + 2 (divide by 4y – 1)

x = (4y + 2) / (4y - 1)

swap x and y

y = (4x + 2) / (4x  - 1)
f^-1(x) = (4x + 2) / (4x - 1)

 

y = log₅(2x - 1) (re-write it as exponential statement)
5^y = (2x - 1) (add 1)
5^y + 1 = 2x (Divide by 2)
(5^y + 1) / 2 = x
swap x and y
y = (5^x + 1) / 2
f^-1(x) = (5^x + 1) / 2

For the first step we simply replace the function with a y.
y=5−9x
Next, replace all the x’s with y’s and all the original y’s with x’s.
x=5−9y
Solve the equation from Step 2 for y.
x=5−9y
9y=5−x
y=(5−x) / 9
Replace y with h⁻¹(x).
h⁻¹(x) = (5-x) / 9