Learn Function Notation, Inputs and Outputs for Digital SAT Math

 

Guide to Functions for Digital SAT Math, GCSE, IGCSE, ACT, and High School Mathematics

Part 1 — Understanding Functions from First Principles

Introduction

Functions form one of the central ideas in mathematics. They appear throughout algebra, coordinate geometry, trigonometry, statistics, calculus, computer science, economics, engineering, physics, finance, and data analysis. A strong understanding of functions makes later topics much easier because many mathematical models are built upon the idea that one quantity depends on another.

Questions involving functions appear regularly in the Digital SAT, PSAT, ACT, GCSE Mathematics, IGCSE Mathematics, and many other secondary school mathematics curricula. Instead of memorising formulas, it is far more useful to understand the reasoning behind functions and how they describe relationships between quantities.

This guide develops every idea gradually. Each new concept builds upon the previous one, allowing you to develop confidence before moving to more advanced function questions.

After studying this chapter, you should be able to:

• Explain the meaning of a function.

• Understand the relationship between an input and an output.

• Read and use function notation correctly.

• Evaluate functions involving positive numbers, negative numbers, fractions and decimals.

• Distinguish between a function and a relation.

• Avoid the most common mistakes made in Digital SAT function questions.

---

What Is a Function?

Many quantities in everyday life depend upon another quantity.

For example,

the total cost of buying notebooks depends upon the number of notebooks purchased.

The distance travelled depends upon the speed and the time.

The amount earned from hourly work depends upon the number of hours worked.

In every case, one quantity changes because another quantity changes.

Mathematics describes these relationships using functions.

A function is simply a rule that matches every permitted input with exactly one output.

The rule never changes unless a different function is defined.

---

Inputs and Outputs

Every function begins with an input.

The function applies a mathematical rule.

The result is called the output.

This relationship can be written as

Input ⟶ Rule ⟶ Output

Consider the following rule.

Multiply the input by 4, then subtract 1.

Input	Rule	Output
1	4(1) − 1	3
3	4(3) − 1	11
6	4(6) − 1	23
10	4(10) − 1	39

Notice an important feature.

Whenever the same input is used, the same output always appears.

This consistency is what makes the rule a function.

---

A Function Describes a Relationship

Suppose a taxi company charges

₹60 as a fixed starting charge

plus

₹18 for every kilometre travelled.

The total fare depends upon the distance travelled.

If x represents the distance travelled in kilometres,

the fare is

60 + 18x

As the distance changes,

the fare changes automatically.

This is another example of a function.

---

Understanding Function Notation

Mathematicians give functions names.

The most common names are

f

g

h

p

or any other convenient letter.

Suppose

f(x) = 4x − 1

The letter f names the function.

The symbol inside the brackets tells us the input.

Many students incorrectly think

f(x)

means

f × x.

It does not.

Instead,

f(5) means

"The value produced by the function when the input equals 5."

---

Evaluating a Function

Suppose

f(x) = 4x − 1

Find

f(7)

Step 1

Replace every x with 7.

f(7)

= 4(7) − 1

Step 2

Multiply.

4 × 7 = 28

Therefore,

f(7)

= 28 − 1

Step 3

Subtract.

28 − 1 = 27

Therefore,

f(7) = 27

---

Example 2

Given

g(x) = 5x + 8

Find

g(12)

Step 1

Replace every x with 12.

g(12)

= 5(12) + 8

Step 2

Multiply.

5 × 12 = 60

Step 3

Add.

60 + 8 = 68

Therefore,

g(12) = 68

---

Example 3

Given

h(x) = 3x² − 2

Find

h(−4)

Step 1

Substitute

x = −4.

h(−4)

= 3(−4)² − 2

Step 2

Calculate the square first.

(−4)² = 16

Therefore,

h(−4)

= 3(16) − 2

Step 3

Multiply.

3 × 16 = 48

Step 4

Subtract.

48 − 2 = 46

Therefore,

h(−4) = 46

Notice that the negative sign disappears after squaring because

(−4) × (−4) = 16.

---

Example 4

Given

p(x) = 6 − 2x

Find

p(3.5)

Step 1

Replace every x with 3.5.

p(3.5)

= 6 − 2(3.5)

Step 2

Multiply.

2 × 3.5 = 7

Step 3

Subtract.

6 − 7 = −1

Therefore,

p(3.5) = −1

Functions may contain integers, fractions or decimals.

The method never changes.

---

Evaluating Functions Containing Fractions

Suppose

f(x) = 3x + 5

Find

f(½)

Step 1

Replace x with ½.

f(½)

= 3(½) + 5

Step 2

Multiply.

3 × ½ = ³⁄₂

Step 3

Write 5 using denominator 2.

5 = ¹⁰⁄₂

Step 4

Add.

³⁄₂ + ¹⁰⁄₂ = ¹³⁄₂

Therefore,

f(½) = ¹³⁄₂

or

6.5

---

Every Occurrence Must Be Replaced

Suppose

q(x) = 2x² + 5x − 7

Find

q(4)

Replace every occurrence of x.

q(4)

= 2(4)² + 5(4) − 7

= 2(16) + 20 − 7

= 32 + 20 − 7

= 45

Many mistakes happen because students replace only the first x.

Always substitute every occurrence.

---

Different Functions Can Produce Different Outputs

Suppose

f(x) = 2x + 1

g(x) = x² + 1

Find

f(4)

and

g(4)

For f(x),

f(4)

= 2(4) + 1

= 9

For g(x),

g(4)

= 4² + 1

= 17

Although both functions use the same input,

their rules are different,

so the outputs are different.

---

What Makes Something a Function?

A function has one important property.

Every permitted input has exactly one output.

If an input produces two different outputs,

the relation is not a function.

For example,

Input	Output
2	7
3	10
4	13
5	16

Every input appears once.

Therefore,

this is a function.

Now consider

Input	Output
2	7
2	11
4	13

Input 2

produces two different outputs.

Therefore,

this is not a function.

---

Common Mistakes

Mistake 1

Thinking

f(x)

means multiplication.

It does not.

It simply names the function.

---

Mistake 2

Replacing only one occurrence of the variable.

Every occurrence of the variable must be substituted.

---

Mistake 3

Ignoring brackets when substituting negative numbers.

Correct:

(−5)²

Incorrect:

−5²

These give different answers.

---

Mistake 4

Ignoring the order of operations.

Always evaluate

1. Brackets

2. Powers

3. Multiplication and division

4. Addition and subtraction

---

Digital SAT Strategy

Most introductory function questions are designed to test careful reading rather than difficult algebra.

Read the function carefully.

Substitute the value accurately.

Complete one operation at a time.

Most calculation mistakes occur because students rush the substitution step.

---

Practice Questions

1. If

f(x) = 7x − 5,

find

f(8).

---

2. If

g(x) = 4x² + 3,

find

g(−2).

---

3. If

h(x) = 15 − 5x,

find

h(4).

---

4. If

p(x) = 6x + 9,

find

p(½).

---

5. Determine whether the following relation is a function.

Input  Output

1    5

2    8

3    11

4    14

---

6. Determine whether the following relation is a function.

Input  Output

3    8

3    10

5    14

---

7. If

q(x) = 2x² − 3x + 4,

find

q(−3).

---

Answers

1. 51

2. 19

3. −5

4. 12

5. Yes

6. No

7. 31

---

Chapter Summary

A function is a mathematical rule that assigns exactly one output to every permitted input. Understanding functions begins with recognising the relationship between inputs, outputs, and the rule connecting them. By learning how to read function notation, substitute values correctly, and identify whether a relation satisfies the definition of a function, you establish the foundation needed for graphing functions, studying domains and ranges, interpreting function values from tables and graphs, and solving more advanced Digital SAT algebra questions.

Click here for more topics/guides for digital sat, gcse etc