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IB MYP 4-5 Maths- Function notation- Study Notes

IB MYP 4-5 Maths- Function notation- Study Notes - New Syllabus

IB MYP 4-5 Maths- Function notation – Study Notes

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  • Function notation

IB MYP 4-5 Maths- Function notation – Study Notes – All topics

Function Notation

Function Notation

A function is a rule that assigns each input exactly one output. Instead of writing “y in terms of x”, we use a special notation:

General Form: \( f(x) \) This is read as “f of x” and means the value of function f when the input is x.

Example: If \( f(x) = 2x + 3 \), then for x = 4, \( f(4) = 2(4) + 3 = 11 \).

Why use Function Notation?

  • Makes writing and calculating outputs easier.
  • Helps represent functions clearly (especially when we have multiple functions).
  • Used for advanced operations like composite and inverse functions.

Steps to Evaluate a Function:

  1. Write the given function rule.
  2. Replace x with the given value.
  3. Simplify the expression to find the output.
  • Domain: The set of all possible input values (x-values).
  • Range: The set of all possible output values (y-values).
  • Codomain: All values that the function could theoretically output.

Example : 

If \( f(x) = 3x – 7 \), find \( f(5) \).

▶️ Answer/Explanation

Substitute x = 5:

\( f(5) = 3(5) – 7 = 15 – 7 = \boxed{8} \)

Example : 

If \( g(x) = x^2 – 4x + 6 \), find \( g(-3) \).

▶️ Answer/Explanation

Substitute x = -3:

\( g(-3) = (-3)^2 – 4(-3) + 6 = 9 + 12 + 6 = \boxed{27} \)

Functions and Their Representations

A function is a rule that assigns exactly one output for each input. In mathematics, we write a function as:

\( y = f(x) \) or in mapping notation: \( f: x \mapsto 2x + 3 \).

Functions vs. Relations

  • A function gives exactly one output for each input.
  • A relation can give more than one output for the same input.

Example:

  • \( f(x) = x^2 \) is a function (every x has one value of y).
  • \( x^2 + y^2 = 25 \) is a relation (a circle: each x corresponds to two y values).

Types of Functions

  • Linear: \( y = mx + c \) (straight line)
  • Quadratic: \( y = ax^2 + bx + c \) (parabola)
  • Exponential: \( y = a \cdot b^x \) (curved growth/decay)
  • Piecewise: Defined by different rules for different intervals

Real-Life Applications

  • Cost function: \( C(x) = 50x + 200 \) (production cost)
  • Speed function: \( s(t) = 20t \) (distance vs. time)
  • Growth function: \( P(t) = P_0 e^{kt} \) (population growth)

Example : 

Write the function \( f(x) = 3x – 2 \) in mapping notation and compute f(5).

▶️ Answer/Explanation

Mapping notation: \( f: x \mapsto 3x – 2 \).

Compute f(5):

\( f(5) = 3(5) – 2 = 15 – 2 = \boxed{13} \)

Example : 

The relation is given by pairs: {(1, 2), (2, 4), (3, 6), (2, 5)}. Is this a function?

▶️ Answer/Explanation

For x = 2, there are two outputs (4 and 5), so it does not satisfy the definition of a function.

Answer: It is a relation but not a function.

Example : 

A taxi company charges a fixed fee of $10 plus $2 per km. Write the cost function and find the cost of 15 km.

▶️ Answer/Explanation

Function: \( C(x) = 10 + 2x \).

For 15 km: \( C(15) = 10 + 2(15) = 10 + 30 = \boxed{\$40} \).

Composite Functions

Composite Functions

A composite function is formed when the output of one function becomes the input of another function. It is written as \( (f \circ g)(x) = f(g(x)) \) and read as “f of g of x”.

 Idea: Apply the inside function first (g), then apply the outside function (f).

Example: If \( f(x) = 2x + 1 \) and \( g(x) = x^2 \), then:

  • \( f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1 \)
  • \( g(f(x)) = g(2x + 1) = (2x + 1)^2 \)

Steps to find a composite function:

  1. Identify the two functions \( f(x) \) and \( g(x) \).
  2. Start with the inner function \( g(x) \) and compute its output for the given input.
  3. Use this output as input for the outer function \( f(x) \).
  4. Simplify the result to get the composite function.

 Note:

  • Understand notation: \( f \circ g \) means f after g (apply g first).
  • Order matters: \( f(g(x)) \neq g(f(x)) \) in most cases.
  • Be able to compute:
    • \( (f \circ g)(x) \) (f after g)
    • \( (g \circ f)(x) \) (g after f)
  • Evaluate composite functions for specific values and as expressions.
  • Understand domain restrictions:
    • The domain of the composite function depends on the inner function.

Example : 

If \( f(x) = 3x – 2 \) and \( g(x) = x^2 \), find \( (f \circ g)(x) \).

▶️ Answer/Explanation

\( f(g(x)) = f(x^2) = 3(x^2) – 2 = 3x^2 – 2 \)

So: \( (f \circ g)(x) = 3x^2 – 2 \)

Example : 

If \( f(x) = 2x + 5 \) and \( g(x) = x – 3 \), find \( (g \circ f)(4) \).

▶️ Answer/Explanation

Compute f(4):

\( f(4) = 2(4) + 5 = 13 \)

Now compute g(f(4)) = g(13):

\( g(13) = 13 – 3 = 10 \)

Final Answer: \( (g \circ f)(4) = 10 \)

Example : 

The cost of producing x units is given by \( g(x) = 50x + 200 \). The tax applied to the cost is \( f(C) = 1.1C \). Find a formula for the total cost after tax in terms of x, and compute it for 10 units.

▶️ Answer/Explanation

Composite function: \( f(g(x)) = 1.1(50x + 200) = 55x + 220 \)

For x = 10:

\( f(g(10)) = 55(10) + 220 = 550 + 220 = \boxed{\$770} \)

Inverse Functions

Inverse Functions

An inverse function reverses the effect of the original function. If a function \( f \) maps \( x \) to \( y \), then its inverse \( f^{-1} \) maps \( y \) back to \( x \).

If \( f(x) = y \), then \( f^{-1}(y) = x \).

Properties:

  • The domain of \( f \) becomes the range of \( f^{-1} \), and vice versa.
  • A function must be one-to-one (each input gives a unique output) to have an inverse function.
  • The graph of \( f \) and \( f^{-1} \) are reflections of each other across the line \( y = x \).

Steps to Find the Inverse of a Function:

  1. Replace \( f(x) \) with \( y \).
  2. Swap \( x \) and \( y \).
  3. Solve for \( y \) in terms of \( x \).
  4. Replace \( y \) with \( f^{-1}(x) \).

Example : 

Find the inverse of \( f(x) = 2x + 5 \).

▶️ Answer/Explanation

Step 1: Write as \( y = 2x + 5 \).

Step 2: Swap x and y: \( x = 2y + 5 \).

Step 3: Solve for y: \( y = \dfrac{x – 5}{2} \).

Final Answer: \( f^{-1}(x) = \dfrac{x – 5}{2} \).

Example : 

A function converts temperature from Celsius to Fahrenheit: \( F = \dfrac{9}{5}C + 32 \). Find the inverse function to convert Fahrenheit to Celsius.

▶️ Answer/Explanation

Step 1: Start with \( F = \dfrac{9}{5}C + 32 \).

Step 2: Swap C and F: \( C = \dfrac{5}{9}(F – 32) \).

Inverse Function:

\( f^{-1}(F) = \dfrac{5}{9}(F – 32) \).

Final Answer: \( f^{-1}(x) = \dfrac{5}{9}(x – 32) \).

Example : 

Find the inverse of \( f(x) = x^2 \) when \( x \ge 0 \).

▶️ Answer/Explanation

Step 1: Write as \( y = x^2 \).

Step 2: Swap x and y: \( x = y^2 \).

Step 3: Solve for y: \( y = \sqrt{x} \).

Step 4: Domain and Range: Since \( x \ge 0 \), the range of the original function becomes the domain of the inverse.

Final Answer: \( f^{-1}(x) = \sqrt{x}, \; x \ge 0 \).

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