Functions - The Basics

In this page, we will look at the different types of relations including functions. The first step to gaining a good understanding of functions is to think about inputs and outputs. Once that is clear, you should be able to deal with domain and range, as well as composite and inverse functions. For the IB examination, questions require you to have a strong conceptual understanding of functions, with questions often posed with graphs. You will be well prepared for that if you work through this page.


Key Concepts

On this page, you should learn to

  • Understand the different types of relation: one-to-one function, many-to-one- function, one-to-many relation, many-to-many relation
  • Use the domain and range of a function
  • Find and use inverse functions \(f^{-1}\)
  • Use composite functions \((f∘g)(x)=f(g(x))\)
  • "Draw" and "Sketch" a graph

Summary

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Test Yourself

Here is a quiz that tests your understanding of the basics of functions, the different types of relation, and also the domain and range of functions:


START QUIZ!

Here is a quiz that tests your understanding of composite and inverse functions


START QUIZ!

Exam-style Questions

Question 1

The function f and g are defined as follows

\(f(x)=3x^2 \ ,\ x\in\mathbb{R}\)

\(g\left(x\right)=\frac{2}{x}-4\ ,x\in\mathbb{R},\ x>0\)

a) Find the range of g

b) Find g-1

c) Solve gf(x) = 2

Hint

Full Solution

 

Question 2

Let f and g be the functionsLet f and g be the functions

\(f(x)=ln(1-2x),x<\frac{1}{2}\)

\(g(x)=\frac{4}{x-2},x\neq2\)

a) Find the exact value of fg(-2)

b) Find f-1(x), stating its domain

c) Show that \(gg(x)=\frac{4x-8}{8-2x}\)

Hint

Full Solution

Question 3

The graph of the function f is plotted below

Use the graph to find values of

a) \(ff(-2)\)

b) \(f^{-1}(0)\)

Let g be the function such that g(x) = 4x + 2

c) Work out fg(1)

d) Draw a sketch of \(y=f^{-1}(x)\)

Hint

Full Solution

 

Question 4

Let \(f(x)=\frac{ax-1}{x-a},\ x\neq\ a\)

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

b) Hence, find ff(b)

c) The graph of y = g(x) is shown below. Find the values of a if fg(3) = a + 5

Hint

Full Solution

 

Question 5

Let the function f be defined by

\(f(x)=\frac{x-k}{2-x},x\neq2\)

a) Find \(f^{-1}(x)\) in terms of k

b) Find k if \(y=f(x)\) and \(y=f^{-1}(x)\) intersect at (-2 , -2) and (3 , 3)

Hint

Full Solution

 

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