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IB Mathematics AA SL Function and their domain range graph Study Notes

IB Mathematics AA SL Function and their domain range graph Study Notes

IB Mathematics AA SL Function and their domain range graph Study Notes Offer a clear explanation of Function and their domain range graph , including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic Function and their domain range graph

Concept of a Function, Domain, Range, and Notation

Function

  • A function is a rule that assigns exactly one output (value) to each input from a given set.
  • Example: \( f(x) = x^2 \) means the function f takes any number \( x \) and returns \( x^2 \).

 Function Notation

  • f(x) means “function f evaluated at x”.
  • Other common notations: \( v(t) \) for velocity as a function of time, \( C(n) \) for cost as a function of quantity \( n \).

 Domain

  • The domain of a function is the set of all input values (x-values) for which the function is defined.
  • Example: For \( f(x) = \frac{1}{x} \), the domain is \( x \ne 0 \).

 Range

  • The range of a function is the set of all possible output values (y-values).
  • Example: For \( f(x) = x^2 \), the range is \( y \ge 0 \).

 Graph of a Function

  • The graph of a function is a visual representation of the input-output pairs.
  • Each point on the graph has the form \( (x, f(x)) \).
  • The graph helps determine properties like increasing/decreasing behavior, maximum/minimum values, and intercepts.

Example

Let \( f(x) = \sqrt{x – 2} \)

Determine the domain and range of the function.

▶️Answer/Explanation

The expression inside the square root must be non-negative:
\( x – 2 \ge 0 \Rightarrow x \ge 1 \)
Domain: \( x \ge 2 \)

Since the square root always returns a non-negative result:
Range: \( y \ge 0 \)

 Function as a Mathematical Model

 Function as a Mathematical Model

  • Functions are used to model real-world relationships between quantities.
  • They help describe how one quantity changes in relation to another.
  • Example: The height of an object thrown upwards can be modeled using a quadratic function of time.

 Common Real-World Function Models

Model TypeExample FunctionContext
Linear\( f(x) = mx + c \)Distance vs Time at constant speed
Quadratic\( f(x) = ax^2 + bx + c \)Height of projectile over time
Exponential\( f(x) = a \cdot b^x \)Bacterial growth, radioactive decay
Logarithmic\( f(x) = \log_b x \)Richter scale for earthquakes

 Key Features of Function Models

  • Domain: The set of allowable inputs (usually time, quantity, etc.)
  • Range: The set of possible outputs (height, cost, speed, etc.)
  • Rate of change: How fast the output changes with respect to input

Example

A car travels at a constant speed of 60 km/h. Let \( d(t) \) represent the distance the car has traveled after \( t \) hours.

Model this situation using a function.

▶️Answer/Explanation

Since distance = speed × time, the function is:
\( d(t) = 60t \)

Domain: \( t \ge 0 \) (time can’t be negative)

Range: \( d \ge 0 \)

This is a linear model showing proportional growth.

Informal Concept of an Inverse Function

Inverse Function

  • An inverse function essentially “undoes” the action of a function.
  • If a function takes an input \( x \) and produces an output \( y \), the inverse function takes \( y \) and returns \( x \).
  • In notation, if \( f(x) = y \), then \( f^{-1}(y) = x \).

Informal Interpretation:

  • The function \( f \) transforms an input to an output.
  • The inverse function \( f^{-1} \) transforms that output back to the input.

Function-Inverse Relationship:

For a function and its inverse:

  • \( f(f^{-1}(x)) = x \)
  • \( f^{-1}(f(x)) = x \)

Example

Let \( f(x) = 2x + 4 \)

Find the inverse \( f^{-1}(x) \), and verify that solving \( f(x) = 10 \) is equivalent to finding \( f^{-1}(10) \).

▶️Answer/Explanation

\( y = 2x + 4 \)

Swap \( x \) and \( y \): \( x = 2y + 4 \)

Solve for \( y \):

Subtract 4: \( x – 4 = 2y \)

Divide by 2: \( y = \frac{x – 4}{2} \)

So, the inverse function is:

\( f^{-1}(x) = \frac{x – 4}{2} \)

Now solve \( f(x) = 10 \):

\( 2x + 4 = 10 \Rightarrow x = 3 \)

Check using the inverse:

\( f^{-1}(10) = \frac{10 – 4}{2} = 3 \)

 So solving \( f(x) = 10 \) is equivalent to finding \( f^{-1}(10) \).

Inverse Functions and Reflection in the Line y = x

Inverse Functions and Reflection in the Line y = x

 Geometric Interpretation:

  • The graph of an inverse function \( f^{-1}(x) \) is the reflection of the graph of \( f(x) \) across the line \( y = x \).
  • This means if a point \( (a, b) \) lies on the graph of \( f(x) \), then the point \( (b, a) \) lies on the graph of \( f^{-1}(x) \).

 Function Inverses Exist Only for One-to-One Functions:

  • A function must be one-to-one (injective) to have an inverse.
  • This means that each output is produced by exactly one input.
  • Horizontal line test: if every horizontal line intersects the graph at most once, the function has an inverse.

Domain and Range Relationship:

  • If \( f(x) \) has domain \( D \) and range \( R \), then: \( f^{-1}(x) \) has domain \( R \) and range \( D \)
  • This reversal of input/output is the essence of an inverse function.

Example

Let \( f(x) = 3x – 5 \). Find \( f^{-1}(x) \), and verify it as a reflection in the line \( y = x \).

▶️Answer/Explanation

\( y = 3x – 5 \)

Swap \( x \) and \( y \): \( x = 3y – 5 \)

Solve for \( y \): 

\( x + 5 = 3y \Rightarrow y = \frac{x + 5}{3} \)

So the inverse is: \( f^{-1}(x) = \frac{x + 5}{3} \)

To verify the reflection:

If \( f(2) = 1 \), then \( f^{-1}(1) = 2 \) 

 Graph both \( f(x) = 3x – 5 \) and \( f^{-1}(x) = \frac{x + 5}{3} \) along with the line \( y = x \) to visually confirm they are reflections of each other.

Inverse Functions in Real-World Contexts

Inverse Functions in Real-World Contexts

Inverse functions are useful in many real-life applications. Two important examples are:

  • Temperature conversions between Celsius and Fahrenheit
  • Currency conversions between different currencies

Temperature Conversion

The formula to convert Celsius to Fahrenheit is:

\( F = \frac{9}{5}C + 32 \)

To convert Fahrenheit back to Celsius (i.e. find the inverse function), 

  1. Start with: \( F = \frac{9}{5}C + 32 \)
  2. Subtract 32: \( F – 32 = \frac{9}{5}C \)
  3. Multiply by 5/9: \( C = \frac{5}{9}(F – 32) \)

This shows that the function converting Celsius to Fahrenheit has an inverse: converting Fahrenheit to Celsius.

Currency Conversion

Suppose the exchange rate is 1 USD = 0.85 EUR.

  • USD to EUR: \( E(x) = 0.85x \), where \( x \) is in USD
  • Inverse (EUR to USD): Solve \( x = 0.85y \) → \( y = \frac{x}{0.85} \)
  • So:\( E^{-1}(x) = \frac{x}{0.85} \)

Example: Convert 30°C  in Fahrenheit?

▶️Answer/Explanation

Use \( F = \frac{9}{5}C + 32 \)

Substitute: \( F = \frac{9}{5}(30) + 32 = 54 + 32 = 86°F \)

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