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

Question 1

Skill question

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

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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).

1.

For the general linear fractional function $f (x) = \frac{p x + q}{r x + s}$ , derive a formula for the inverse function $f^{- 1} (x)$ .

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2.

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

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Question 3

Skill question

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

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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.

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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 $f^{- 1} (f (x)) = x$ .

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Question 6

Skill question

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

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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.

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Question 8

Skill question

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

(a) Find $f^{- 1} (x)$ .

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

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Question 9

Skill question

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

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Question 10

Skill question

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

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Question 11

Skill question

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

1.

Find $f^{- 1} (x)$ .

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2.

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

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IB Mathematics Applications & Interpretation Question Type 3: Finding the Inverse Function of Simple Functions Exercises