Functional Notation f(x)

Reading and using function notation

What is Function Notation?

Function notation is a way to name and write functions. Instead of writing \(y = 2x + 3\), we write \(f(x) = 2x + 3\).

The notation \(f(x)\) is read as "f of x" and represents the output of the function when we input x.

Traditional Form:

\(y = 2x + 3\)

Function Notation:

\(f(x) = 2x + 3\)

\(f(x)\) means "the value of f at x" or "f of x"

Understanding the Notation

Breaking Down f(x)

In the expression \(f(x) = 2x + 3\):

f is the name of the function
x is the input variable (inside the parentheses)
2x + 3 is the rule that tells us what to do with x
f(x) represents the output value

Note: We can use any letter for functions: \(g(x)\), \(h(x)\), \(p(x)\), etc.

Evaluating Functions

Example 1: Finding f(2)

Given \(f(x) = 3x - 5\), find \(f(2)\)

1 Replace x with 2:

\[f(2) = 3(2) - 5\]

2 Calculate:

\[f(2) = 6 - 5 = 1\]

Answer: \(f(2) = 1\)

Example 2: Evaluating with Different Inputs

Given \(g(x) = x^2 + 2x - 1\), find:

(a) g(3)

\[g(3) = (3)^2 + 2(3) - 1 = 9 + 6 - 1 = 14\]

(b) g(-2)

\[g(-2) = (-2)^2 + 2(-2) - 1 = 4 - 4 - 1 = -1\]

(c) g(0)

\[g(0) = (0)^2 + 2(0) - 1 = 0 + 0 - 1 = -1\]

Example 3: Evaluating with Expressions

Given \(f(x) = 2x + 1\), find \(f(a + 3)\)

1 Replace every x with (a + 3):

\[f(a + 3) = 2(a + 3) + 1\]

2 Expand and simplify:

\[f(a + 3) = 2a + 6 + 1 = 2a + 7\]

Interactive Function Evaluator

Calculate f(x)

Function:

Input value x =

Result

\(f(x) = 2x + 3\)

\(f(2) = 7\)

Table of Values

x	-2	-1	0	1	2	3
f(x)	-1	1	3	5	7	9

Finding x When f(x) is Given

Example 4: Solving for x

Given \(f(x) = 4x - 7\), find x when \(f(x) = 13\)

1 Set the function equal to 13:

\[4x - 7 = 13\]

2 Solve for x:

\[4x = 13 + 7 = 20\]

\[x = 5\]

Verify: \(f(5) = 4(5) - 7 = 20 - 7 = 13\) ✓

Multiple Functions

Example 5: Working with Two Functions

Given \(f(x) = x + 2\) and \(g(x) = 3x\), find:

(a) f(4) + g(2)

\[f(4) = 4 + 2 = 6\] \[g(2) = 3(2) = 6\] \[f(4) + g(2) = 6 + 6 = 12\]

(b) f(g(1))

First find \(g(1) = 3(1) = 3\) Then find \(f(3) = 3 + 2 = 5\) \[f(g(1)) = 5\]

Practice Problems

Question 1: If \(f(x) = 5x - 2\), find \(f(3)\)

Show Solution

\(f(3) = 5(3) - 2 = 15 - 2 = 13\)

Question 2: If \(g(x) = x^2 - 3x + 2\), find \(g(-1)\)

Show Solution

\(g(-1) = (-1)^2 - 3(-1) + 2\)

\(= 1 + 3 + 2 = 6\)

Question 3: If \(h(x) = 2x + 5\), find x when \(h(x) = 15\)

Show Solution

\(2x + 5 = 15\)

\(2x = 10\)

\(x = 5\)

Question 4: If \(f(x) = 3x - 1\), find \(f(2a)\)

Show Solution

Replace x with 2a:

\(f(2a) = 3(2a) - 1 = 6a - 1\)

Question 5: Given \(f(x) = x + 1\) and \(g(x) = x^2\), find \(g(f(2))\)

Show Solution

First: \(f(2) = 2 + 1 = 3\)

Then: \(g(3) = 3^2 = 9\)

\(g(f(2)) = 9\)

CSEC Exam Tips

\(f(x)\) and y are interchangeable - they both represent the output
Always substitute carefully, especially with negative numbers
Use brackets when substituting to avoid sign errors
When finding x from f(x), set up and solve an equation
Read the question carefully - it might ask for f(x) OR x