Use function notation, domain, range, and inverse ideas

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Functions

IB Mathematics: Analysis and Approaches SL — Study topic SL 2.2

A function is a rule that assigns exactly one output to each allowed input. Before working with a function, ask which inputs are allowed and which outputs can result. These sets are its domain and range. Function notation gives a compact way to describe inputs and outputs; inverse notation describes a rule that reverses the input-output relationship. You will use these ideas algebraically, numerically, graphically, and in a simple context.

What you will learn

Interpret and use function notation to find outputs and solve for inputs.
Identify a function's domain and range from an algebraic rule or a context.
Find an inverse function when the original function has a suitable domain.
Connect inverse functions to reflected graphs and exchanged input-output pairs.

1. Prior knowledge and function notation

An expression such as

3x+2

describes a calculation. The notation

f(x)=3x+2

names that rule:

f

is the function, and

x

is its input. The value

f(4)

means the output when the input is

4

, not multiplication of

f

by

4

. Substitute the input wherever the variable appears, then simplify.

A function must give one output for each allowed input. For example, if

f(x)=3x+2

, then

f(4)=14

. If you know an output and need its input, set the rule equal to that output and solve. Keeping track of which quantity is input and which is output is useful in both calculations and contexts.

f(x)=3x+2

The letter inside the brackets is the input; the result is the output.
To solve $f(x)=k$ , set the function rule equal to $k$ and solve for $x$ .

2. Domain and range

The domain is the set of inputs allowed for a function. The range is the set of outputs the function actually produces from those inputs. A rule alone may not determine the intended domain: a question or context can impose additional restrictions. State such restrictions clearly.

For a real-valued rule involving a square root, the expression under the square root must be non-negative. In a practical setting, quantities may also be restricted: a time measured after an event cannot be negative, for example. These restrictions affect the domain and therefore can also affect the range.

A table of input-output pairs makes the relationship numerical. For a graph, inputs appear along the horizontal axis and outputs along the vertical axis. The graph helps you see which input and output values occur, but use the rule and stated restrictions to justify exact endpoints. When using interval notation, a square bracket includes an endpoint and a round bracket excludes it.

Domain describes allowed inputs; range describes resulting outputs.
Check the rule and the context before stating either set.
Changing the domain can change the range.

3. Inverse functions and representations

An inverse function reverses the input-output pairs of the original function. If a function takes input

a

to output

b

, its inverse takes input

b

to output

a

. The notation

f^{-1}(x)

means the inverse function; it does not mean the reciprocal

1/f(x)

.

To find an inverse, write the output as a variable, interchange the input and output, and solve for the new output. Then write the result using inverse notation. The domain of the inverse is the range of the original function, and its range is the domain of the original function. This exchange is a useful check.

An inverse must still be a function: each input to the inverse must produce only one output. A graph that gives two different outputs for one input cannot itself be the graph of a function. If a rule does not reverse uniquely on its full domain, a stated restriction may make its outputs correspond to inputs one-to-one. Do not silently change a domain; use only a restriction given or justified in the question.

Graphically, the graphs of a function and its inverse are reflections of each other across the line

y=x

. Numerically, each ordered pair

(a,b)

becomes

(b,a)

. A graphing calculator can help compare the two graphs or inspect possible domain and range, but it does not replace the algebraic steps or the need to state restrictions.

f(a)=b\iff f^{-1}(b)=a

The inverse reverses input and output; it is not the reciprocal.
The original range becomes the inverse domain, and the original domain becomes the inverse range.
For a graph, inverse pairs are reflected across $y=x$ .

Worked example

Evaluate a function and find its domain and range

Let

f(x)=3x-2

for

-1\leq x\leq 4

. Find

f(3)

, then state the domain and range.

Substitute the input
Replace each occurrence of $x$ with $3$ . The input is allowed because it lies between the stated endpoints.
$f(3)=3(3)-2=7$
Read the domain
The question allows every input from $-1$ to $4$ , including both endpoints, so use square brackets.
$-1≤ x≤ 4$
Find the output endpoints
The rule is linear and increases as $x$ increases, so its smallest and largest outputs occur at the domain endpoints.
$f(-1)=-5, f(4)=10$

Answer:

f(3)=7

. The domain is

[-1,4]

and the range is

[-5,10]

.

Check: The endpoint outputs are included because the input endpoints are included. Every intermediate input produces an output between

-5

and

10

.

Worked example

Find an inverse with a restricted domain

Let

g(x)=(x-2)^2+1

with domain

x\geq 2

. Find

g^{-1}(x)

and state its domain and range.

Write an output variable
Use $y$ for the output so that interchanging input and output is clear.
$y=(x-2)^2+1$
Interchange input and output
Reverse the input-output relationship. The original restriction $x\geq2$ ensures the inverse uses the non-negative square root.
$x=(y-2)^2+1$
Solve for the new output
Subtract $1$ and take the non-negative square root, since the original input was at least $2$ . This selects the branch consistent with the stated domain.
$y=2+\sqrt{x-1}$
Exchange domain and range
The original function has minimum output $1$ and no upper limit. Its inverse therefore accepts inputs from $1$ upward and returns values from $2$ upward.
$g^{-1}(x)=2+\sqrt{x-1}, x\geq1$

Answer:

g^{-1}(x)=2+\sqrt{x-1}

, with domain

x\geq1

and range

y\geq2

.

Check: For example,

g(4)=5

and

g^{-1}(5)=4

, so the input-output pair is reversed correctly.

Worked example

Interpret an inverse in a context

A delivery service estimates its fee in dollars using

C(t)=5t+8

, where

t

is the distance in kilometres and

t\geq0

. Find the distance that corresponds to a fee of CAD 38, and write the inverse rule.

Set the fee to the given amount
The fee is the output of $C$ . Substitute the given fee and solve for the distance input.
$5t+8=38$
Solve for distance
Subtract the fixed fee and divide by the rate per kilometre. The result respects the stated non-negative distance domain.
$t=6$
Reverse the rule
Write the fee as the input and solve for distance. The inverse accepts fees of at least CAD 8, corresponding to distances of at least zero.
$C^{-1}(x)=\frac{x-8}{5}, x\geq8$

Answer: The fee of CAD 38 corresponds to a distance of

6

km. The inverse rule is

C^{-1}(x)=\frac{x-8}{5}

for

x\geq8

.

Check: Substituting

6

km into the original rule gives

5(6)+8=38

, so the context and inverse agree.

Common mistakes and how to avoid them

Treating

f(3)

as

f

multiplied by

3

.

Correction: Read the bracketed value as the input to the function, then substitute it into the rule.

Giving a range without considering the stated domain.

Correction: Find outputs only from allowed inputs; a domain restriction can change the range.

Writing

1/f(x)

for the inverse.

Correction: An inverse reverses input and output pairs. Use reciprocal notation only when division by a function value is intended.

Taking both square-root signs when a restricted domain selects one.

Correction: Use the original domain to choose the output branch that reverses the given function.

Lesson summary

Function notation names a rule and identifies its input and output.
The domain is the set of allowed inputs; the range is the set of resulting outputs.
An inverse reverses input-output pairs, so the original domain and range exchange roles.
Check an inverse algebraically with an input-output pair and, when useful, graphically by reflection in $y=x$ .

Check your understanding

Question 1

If

h(x)=2x+5

, what is

h(4)

?

13
18
7
9

Show answer and explanation

13

Substitute

4

for

x

:

2(4)+5=13

.

Question 2

The function

p(x)=x^2

has domain

x\geq0

. What is its range?

All real numbers
$y\geq0$
$y>0$
$x\geq0$

Show answer and explanation

y\geq0

Squaring a non-negative input gives an output at least zero, and zero is attained when

x=0

.

Question 3

If

q(2)=7

, which statement must be true for the inverse?

$q^{-1}(2)=7$
$q^{-1}(7)=2$
$q^{-1}(7)=7$
$q^{-1}(2)=2$

Show answer and explanation

q^{-1}(7)=2

The inverse reverses the input-output pair, so the input

7

produces output

2

.

Key terms

Function: A rule that assigns exactly one output to each allowed input.
Domain: The set of inputs for which a function is defined.
Range: The set of outputs produced by a function over its stated domain.
Inverse function: A function that reverses the input-output pairs of the original function.

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