Understanding Functions

⚙️ Understanding Functions

Part 1 of 5 — What Is a Function?

Topics in This Part

🔑 Key Concept: A function is a rule that takes each input and gives back exactly one output. Same input in → same single output out, every time.

The Input–Output Machine

Picture a function as a machine. You drop an input in the top, the machine applies its rule, and exactly one output drops out the bottom.

$\text{input} \;\longrightarrow\; \boxed{\text{rule}} \;\longrightarrow\; \text{output}$

For example, a "double it" machine:

The inputs together are called the domain. The outputs together are called the range.

💡 You already use functions every day. "Hours worked" → "pay earned" is a function. So is "number of tacos" → "total cost."

The One Rule That Defines a Function

Here is the single test that decides everything:

🔑 Function Rule: Each input is allowed exactly one output. An input may never point to two different outputs.

It is perfectly fine for two different inputs to share the same output:

The first relation is fine even though $1$ and $2$ both output $4$. The problem in the second is that the same input $1$ produces two different outputs.

⚠️ Watch the direction. Repeated outputs are allowed. Repeated inputs with different outputs are NOT.

Concept Check 🎯

Function or Not? 🔽

For each table of $(x, y)$ pairs, decide whether $y$ is a function of $x$.

Run the Machine 🧮

A "double it" function uses the rule $y = 2x$ — it sends each input to twice itself.

1) Input $4$ gives output $\,?$ 2) Input $7$ gives output $\,?$ 3) Which input gives the output $20$? Input $= \,?$

You've Got the Definition

You can now state the one rule that makes a relation a function: one output per input. You can test a table by scanning the input column for a repeat that points to different outputs.

In Part 2 we'll see the four ways the same function can be written — table, mapping diagram, graph, and equation — and meet the famous vertical line test.

⚙️ Understanding Functions

Part 2 of 5 — Four Ways to Show a Function

🔑 The Idea: The same function can be shown as a table, a mapping diagram, a graph, or an equation. They all carry the same input → output information.

The Four Representations

Here is the rule "add 1" shown four ways:

1) Table

2) Mapping diagram — arrows from each input to its output:

$0 \to 1, \quad 1 \to 2, \quad 2 \to 3$

3) Equation (rule):

$y = x + 1$

4) Graph — the points $(0,1)$, $(1,2)$, $(2,3)$ plotted on a coordinate grid.

💡 Whenever you switch representations, the pairs never change. The point $(2,3)$, the table row $2 \to 3$, and "plug $x=2$ into $y=x+1$" all say the same thing.

Concept Check 🎯

The Vertical Line Test

On a graph, every input $x$ corresponds to a vertical line. If that vertical line hits the graph more than once, then that single input has two outputs — so the graph is not a function.

🔑 Vertical Line Test (VLT): If any vertical line crosses the graph more than once, the graph is not a function. If every vertical line crosses at most once, it is a function.

⚠️ A parabola passes the VLT even though it has a repeated output (e.g. $y=4$ at $x=2$ and $x=-2$). Remember: repeated outputs are allowed.

Apply the Vertical Line Test 🔽

Decide whether each graph passes the vertical line test (and is therefore a function).

Translate Between Representations 🧮

The function is given by the rule $y = 3x$.

1) Complete the table: when $x = 2$, $y = \,?$ 2) When $x = 5$, $y = \,?$ 3) The graph passes through the point $(4, \,?)$ — enter the missing $y$.

⚙️ Understanding Functions

Part 3 of 5 — Function Rules & Evaluating

🔑 Why it matters: A function's rule (its equation) lets you find the output for any input — even inputs not in the table — by substituting and simplifying.

Evaluating a Function Rule

To evaluate a function, replace $x$ with the input value and simplify.

Example: $y = 2x + 5$

Work the order of operations carefully — multiply before you add.

Example: $y = x^2 - 4$ at $x = 3$

$y = (3)^2 - 4 = 9 - 4 = 5$

⚠️ Square the input before subtracting. $3^2 = 9$, not $3 \cdot 2 = 6$.

Concept Check 🎯

Evaluate the Rule 🧮

Use the rule $y = 3x - 4$.

1) $x = 0 \;\Rightarrow\; y = \,?$ 2) $x = 5 \;\Rightarrow\; y = \,?$ 3) $x = -2 \;\Rightarrow\; y = \,?$

Reading a Rule From Words

Many word problems hide a function. Translate the words into a rule.

Example: Taco truck

Tacos cost $2 each plus a $3 flat delivery fee. The total cost $C$ for $t$ tacos is:

$C = 2t + 3$

• The $2 per taco is the number multiplied by the input (the rate).
• The $3 fee is added once no matter what (the starting value).

So $5$ tacos cost $C = 2(5) + 3 = 13$, giving $13.

💡 "Per," "each," and "every" signal the number you multiply by. "Flat fee," "starting," and "one-time" signal the number you add.

Build the Rule 🔽

A gym charges a $20 sign-up fee plus $15 each month. Let $m$ be the number of months and $C$ the total cost.

⚙️ Understanding Functions

Part 4 of 5 — Rate of Change, Initial Value & Linear vs. Nonlinear

🔑 Big Idea: A linear function changes by the same amount every time the input goes up by $1$. That steady change is the rate of change (slope), and the output at $x=0$ is the initial value.

Rate of Change & Initial Value

For a linear function written $y = mx + b$:

• $m$ is the rate of change (slope) — how much $y$ changes each time $x$ increases by $1$.
• $b$ is the initial value — the output when $x = 0$ (the $y$-intercept).

$\text{rate of change} = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}$

Example: From a table

Each time $x$ goes up by $1$, $y$ goes up by $3$, so the rate of change is $3$. The initial value (at $x=0$) is $4$. The rule is $y = 3x + 4$.

💡 To find the rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$, use $\dfrac{y_2 - y_1}{x_2 - x_1}$.

Find Rate of Change & Initial Value 🧮

A linear function passes through these points:

1) Rate of change (change in $y$ per $+1$ in $x$) $= \,?$ 2) Initial value (output at $x = 0$) $= \,?$ 3) Using $y = mx + b$, what is the output at $x = 5$? $\,?$

Linear vs. Nonlinear

A function is linear if it has a constant rate of change — its graph is a straight line and its equation can be written $y = mx + b$ (no $x^2$, no $x$ in a denominator, no $x$ under a root).

Check $y = x^2$ with a table

The jumps are $+1, +3, +5$ — not constant — so $y = x^2$ is nonlinear.

⚠️ A table is linear only if the output changes by the same amount for each equal step in $x$.

Concept Check 🎯

Comparing Two Functions

You can compare functions even when they're written differently — pull the rate of change and initial value from each.

Example

Function A (equation): $y = 4x + 1$. Rate of change $= 4$, initial value $= 1$.

Function B (table):

Function B has rate of change $2$ and initial value $3$.

So A grows faster (rate $4 > 2$), but B starts higher (initial value $3 > 1$).

💡 To compare, get every function into the same language: how fast it grows (rate of change) and where it starts (initial value).

Compare the Functions 🔽

Function A: $y = 2x + 6$. Function B: a line through $(0, 2)$ and $(1, 7)$.

⚙️ Understanding Functions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) define a function and test relations, (2) move between table, mapping, graph, and equation, (3) evaluate and build rules, and (4) find rate of change, initial value, and tell linear from nonlinear. Let's put it together.

Quick Reference

⚠️ Repeated outputs are allowed. Repeated inputs with different outputs are not.

Mixed Practice 🎯

Mixed Drill 🧮

A line passes through $(0, 5)$ and $(2, 11)$.

1) Rate of change $= \dfrac{11 - 5}{2 - 0} = \,?$ 2) Initial value (output at $x=0$) $= \,?$ 3) Write the output at $x = 4$ using $y = mx + b$: $\,?$

Exit Quiz ✅

Answer all three to finish the lesson.