Exercises for Question Type 3: Finding the inverse function of simple functions - IB

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Exercises for Question Type 3: Finding the inverse function of simple functions - IB

Written by IB examiners

Question 1

Skill question

Determine the inverse of f (x) = \frac{3 x - 4}{5 x + 2} .

[3]

Question 2

Skill question

This question explores the properties of linear fractional functions, specifically the derivation of their inverse functions and the conditions required for them to be one-to-one (invertible).

(a)

For the general linear fractional function , derive a formula for the inverse function .

[4]

(b)

State the condition on the constants p, q, r, s for the function f (x) to be invertible.

[1]

Question 3

Skill question

Find the inverse of f (x) = \frac{x - 2}{x + 3} and state its domain and range.

[5]

Question 4

Skill question

For f (x) = \frac{2 x + 1}{x - k}, determine the value of k for which f is its own inverse.

[4]

Question 5

Skill question

The question asks for a formal derivation and algebraic verification of the inverse function property for a given rational function. It assesses skills in algebraic manipulation and composition of functions.

Given f (x) = \frac{5 x - 3}{2 x + 1}, show that f^{- 1} (f (x)) = x .

[6]

Question 6

Skill question

Find f^{- 1} (x) for f (x) = \frac{4 x + 1}{2 x - 3} .

[3]

Question 7

Skill question

Let f (x) = \frac{x + b}{b x - 1} where b x \neq = 1 . Determine all real values of b for which f is its own inverse.

[6]

Question 8

Skill question

For f (x) = \frac{3 x + 7}{x - 5},

(a) Find f^{- 1} (x) . (b) Show that f (f^{- 1} (x)) = x .

[6]

Question 9

Skill question

Let f (x) = \frac{2 x - 1}{3 x + 4} . Prove that f (f^{- 1} (x)) = x and f^{- 1} (f (x)) = x .

[8]

Question 10

Skill question

Find the inverse function of f (x) = \frac{2 x + 5}{3 x} .

[3]

Question 11

Skill question

Given parameters a, b, c, define the function f (x) = \frac{a}{x - b} + c .

(a)

Find f^{- 1} (x) .

[3]

(b)

Show that f (f^{- 1} (x)) = x .

[2]