Inverse Functions Practice Questions

An inverse function (or anti-function) is a function that “reverses” another function: if the function f applied to an input x, and gives a result of y, then the inverse function g, applied to y,  gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x

1. Find the inverse function of the function f(x) = 3x + 3.

a. (3x – 1) / 3
b. (x + 1) / 3
c. (x – 1) / 3
d. (x – 3) / 3

2. Find the inverse function of the function f(x) = (5x – 2) / 4.

a. (4x – 1) / 5
b. (x + 4) / 5
c. (4x + 2) / 5
d. (4x – 2) / 5

3. Find f ^-1(1/2)  if  f(x) = 1 – x.

a. 1
b. 1/2
c. 1/3
d. 1/4

4. If f(x) = 5x and g(x) = 7 – 2x, find (f – g)^-1 (0).

a. 1
b. 2
c. 3
d. 4

5. If f ^-1 (x) = 2x, find f(x).

a. x
b. 2x
c. x/2
d. x/3

1. D (x-3)/3

f(x) = 3x + 3

f ^-1(f(x)) = x

f ^-1 (3x + 3) = x

3x + 3 = t

3x = t – 3

x = (t – 3) / 3

 f ^-1(t) =  (t – 3) / 3

 f ^-1(x) = (x – 3) / 3

 

2. C. (4x + 2) / 5

f(x) = (5x – 2) / 4

f ^-1(f(x)) = x

f ^-1((5x – 2) / 4) = x

(5x – 2) / 4 = t

5x – 2 = 4t

5x = 4t + 2

x = (4t + 2) / 5

 f ^-1(t) = (4t + 2) / 5

f ^-1(x) = (4x + 2) / 5

 

3. B. 1/2

f(x) = 1 – x

f ^-1(1-x) = x

1 – x = t

x = 1 – t

 f ^-1(t) = 1 – t

 f ^-1(x) = 1 – x

 f ^-1(1/2) = 1 – 1/2 = 1/2

 

4. A. 1

f(x) = 5x

g(x) = 7 – 2x

 f(x) – g(x) = 5x – (7 – 2x) = 5x – 7 + 2x = 7x – 7

(f(x) – g(x))^-1(f(x) – g(x)) = x

(f(x) – g(x))^-1(7x – 7) = x

7x – 7 = t

7x = t + 7

x = (t + 7) / 7

(f(t) – g(t))^-1(t) = (t + 7) / 7

(f(x) – g(x))^-1(x) = (x + 7) / 7

(f(x) – g(x))^-1(0) = (0 + 7)/7 = 1

 

5. C. x/2

f ^-1(f(x))=x       We treat f(x) as a variable, so instead of 2x we have 2f(x).

2f(x) = x

 f(x) = x/2