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Functions in Algebra: Inputs, Outputs, and Function Notation

Capítulo 1

Estimated reading time: 5 minutes

What a Function Is: One Input, Exactly One Output

A function is a rule (or process) that assigns each input to exactly one output. You can think of it like a machine: you put something in, and the machine produces one result.

We often name a function with a letter such as f. If the input is x, the output is written as f(x), read “f of x.” This notation emphasizes the input–output idea: x goes in, f(x) comes out.

Function Notation as an Input–Output Statement

x is the input.
f(x) is the output produced by the rule.
The statement f(3) = 10 means: “When the input is 3, the output is 10.”

Example (verbal rule): “Take the input, double it, then add 1.” If the function is named f, then f(x) = 2x + 1. For input 4: f(4) = 2(4) + 1 = 9.

(1) Key Vocabulary

Input: the value you choose to put into the function (often x).
Output: the value the function produces from that input (often y or f(x)).
Independent variable: the input variable; it can vary freely (commonly x).
Dependent variable: the output variable; it depends on the input (commonly y or f(x)).
Domain: the set of allowed inputs.
Range: the set of possible outputs.

In many algebra settings, you will see y = f(x). This means y is the output of the function f when the input is x.

Function Notation and Mapping Diagrams

A mapping diagram shows inputs pointing to outputs. Each input must point to exactly one output for the relationship to be a function.

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Inputs (x)        Outputs (f(x))
  1   --------->      3
  2   --------->      5
  3   --------->      7

This diagram represents the rule f(x) = 2x + 1 for inputs 1, 2, and 3.

(2) Multiple Representations of the Same Function

The same function can be represented in different ways: a verbal rule, an equation, a table, or a mapping diagram. Being able to move between these representations is a key algebra skill.

Example Function: “Square the input, then subtract 1”

Verbal rule: “Square the input, then subtract 1.”

Equation (function notation): f(x) = x^2 - 1

Table:

Input x	Output f(x) = x^2 − 1
-2	3
-1	0
0	-1
1	0
2	3

Mapping idea (for selected inputs):

-2 -> 3
-1 -> 0
 0 -> -1
 1 -> 0
 2 -> 3

Step-by-Step: Evaluating a Function Value

To find an output like f(5) when f(x) = x^2 - 1:

Replace x with 5: f(5) = 5^2 - 1
Compute: f(5) = 25 - 1 = 24

To find f(-3):

Substitute: f(-3) = (-3)^2 - 1
Square first: (-3)^2 = 9
Finish: f(-3) = 9 - 1 = 8

(3) Deciding Whether a Relationship Is a Function

The rule “each input has exactly one output” is the test. A relationship is not a function if any single input is paired with two or more different outputs.

Example A: x → x² (This IS a function)

Rule: input x, output x^2.

If x = 2, output is 4.
If x = -2, output is also 4.

It is okay for different inputs to share the same output. What matters is that each single input produces only one output.

Example B: x → ±√x (This is NOT a function)

Rule: input x, output is “plus or minus the square root of x.”

Try input x = 9:

+√9 = 3
-√9 = -3

The same input 9 leads to two outputs (3 and −3), so this relationship is not a function.

Real-World Style Checks

Person → birthdate: function (each person has exactly one birthdate).
Birthdate → person: not a function (many people can share the same birthdate).
Student → student ID number: function (each student has one ID).
Student → phone number: may not be a function (a student could have multiple phone numbers, or none).

Table Check: Is It a Function?

A table represents a function if no input value appears with two different outputs.

x	y
1	4
2	5
2	7

This is not a function because the input 2 maps to both 5 and 7.

(4) Guided Practice: Translating Between Words and Notation

A. Words → Function Notation

1) Verbal rule: “Add 6 to the input.”

Let the input be x.
Output is x + 6.
Function: f(x) = x + 6

2) Verbal rule: “Multiply the input by 3, then subtract 2.”

Multiply by 3: 3x
Subtract 2: 3x - 2
Function: g(x) = 3x - 2

3) Verbal rule: “Take half the input, then square the result.”

Half the input: x/2
Square: (x/2)^2
Function: h(x) = (x/2)^2

B. Function Notation → Words

1) Expression: f(x) = x^2 + 5

Words: “Square the input, then add 5.”

2) Expression: p(x) = 4 - x

Words: “Start with 4, then subtract the input.”

3) Expression: m(t) = 2t + 10

Words: “Multiply the input by 2, then add 10.”

C. Using f(x) Statements to Describe Inputs and Outputs

Suppose f(x) = 2x - 3.

f(0) = 2(0) - 3 = -3 (input 0 gives output −3)
f(4) = 2(4) - 3 = 5 (input 4 gives output 5)
If f(x) = 9, then solve 2x - 3 = 9 to find the input: 2x = 12, so x = 6. That means f(6) = 9.

Check for Understanding

In the statement f(7) = 20, identify the input and the output.
Let g(x) = x^2 - 4. Find g(3) and g(-3).
Write a function in notation for the rule: “Subtract 8 from the input, then multiply by 5.”
Decide whether the relationship is a function: “Input x maps to output ±x.” Explain using the one-input/one-output rule.
A table shows pairs (1, 2), (2, 2), (3, 5), (2, 2). Is it a function? Why?
Translate into words: h(x) = (x + 1)/3.
If f(x) = 4x + 1, write an equation you would solve to find the input that produces output 29.

Now answer the exercise about the content:

Which statement correctly explains why the rule x → x² is a function, while the rule x → ±√x is not?

You are right! Congratulations, now go to the next page

You missed! Try again.

A function assigns each input to exactly one output. The rule x → x² gives one result per input, but x → ±√x can produce two outputs for a single input (e.g., 9 → 3 and −3).