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Grade 11 Precalculus · Lesson 2.1

Functions

One visual journey from input and output to piecewise rules, difference quotients, and domain decisions.

8 learning scenesInteractiveSame lesson material

Explanation

Explore

The Function Idea

Interactive lab

One function, four representations

Each input has exactly one output. Explore the rule, then calculate the output for your chosen input.

Input x = 3 →

f(x)=2x+1

→ one output

Choose an input value: 3

Predict f(3).

Double the input, then add 1.

Same function, different form.

Function ✓

Every input has exactly one outgoing arrow.

Not a function ✕

Input 2 has two outgoing arrows, so it has two outputs.

Worked example

Function or not?

Input -1 maps to 4, input 0 maps to 2, and input 3 maps to 4.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Check the inputs

Does any input have more than one output?

Your turn

Which statement correctly describes a function?

Each input has exactly one outputEach output must have exactly one inputInputs cannot repeatOutputs cannot repeat

Explanation

Connect

Connect the Representations

Explanation

You already explored the four forms above. Now connect them.

A table row $(x,f(x))$ becomes a point with the same coordinates on the graph.

Table row

(2,5)

→

Ordered pair

(x,f(x))=(2,5)

→

Graph

same point

Your turn

Question 1 of 2

A table lists values of

x

and

f(x)

. Which representation is being used?

VerbalAlgebraicGraphicalNumerical

Explanation

See the sets

Domain & Range

Domain

The input set

-204

Range

The output set

Visual example: (-2, 3), (0, 5), (4, 3)

Domain = -2, 0, 4 · Range = 3, 5. A repeated output is listed once in a set.

Read domain and range from new ordered pairs

Find the domain and range of

\{(-3,2),(1,6),(5,2)\}

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Read the inputs

Which coordinates form the domain?

Quick Check

Check the idea before moving on.

For the relation with ordered pairs

(-4,7)

(2,9)

, and

(6,7)

, what is the range?

Explanation

Substitute

Function Notation & Evaluating

Interactive lab

Evaluate a quadratic function

Choose an input. Substitute it into every x, keeping negative inputs inside parentheses.

f(x)=3x^2+x-5

Substitute into the rule: $f(-2)=3(-2)^2+(-2)-5$

What is f(-2)?

Explanation

Treat the input as one complete object.

Replace every occurrence of x. For a negative or symbolic input, parentheses protect the substitution.

Worked example

f(a-2)=(a-2)^2-3(a-2)+2

Simplify

f(a-2)=a^2-7a+12

Worked example

Worked example · numerical input

1 · Input

-3

2 · Substitute

2(-3)^2-4(-3)+1

3 · Simplify

31

Numerical input

Evaluate

p(-3)

for

p(x)=2x^2-4x+1

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Substitute carefully

Which substitution is correct?

Worked example · symbolic input

Function rule

q(x)=x^2+2x-4

1 · Input

b+1

2 · Substitute

(b+1)^2+2(b+1)-4

3 · Simplify

b^2+4b-1

Symbolic input

For

q(x)=x^2+2x-4

, find

q(b+1)

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Replace x everywhere

What replaces $x$ ?

Your turn

g(x)=2x^2-1

, what is the correct first substitution for

g(-a)

2a-1

2(-a)-1

2(-a)^2-1

-2a^2-1

Explanation

Choose the rule

Piecewise Functions & Breakpoints

The function

f(x)=\begin{cases}x+3,&x<-2\\x^2,&-2\le x\le2\\5-x,&x>2\end{cases}

Read each rule together with the interval that owns it.

Interactive lab

Choose the piecewise rule

Choose the interval first, then calculate with just that branch. Test −2 and 2 carefully.

f(x)=\begin{cases}x+3,&x<-2\\x^2,&-2\le x\le2\\5-x,&x>2\end{cases}

Input x = -3

Which branch owns x = -3?

Calculate f(-3).

Key takeaway

Mr. Azzam's routine

Number line → interval → rule → substitute.

At a breakpoint, use the inequality that includes equality.

Piecewise graph · visible immediately

Match each colored piece to its interval, then use the Desmos calculator below to explore it yourself.

y=x+3\quad (x<-2)

y=x^2\quad (-2\le x\le2)

y=5-x\quad (x>2)

Worked example

New example · different function

g(x)=\begin{cases}2x-1,&x<0\\x^2+1,&0\le x\le3\\7-x,&x>3\end{cases}

Use this new function for the worked examples below.

At and around a breakpoint

For the new piecewise function, find

g(-2)

g(0)

, and

g(2)

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Left of 0

Which rule owns $x=-2$ ?

The other breakpoint

For the same new function, find

g(3)

and

g(5)

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Exactly at 3

Which rule includes $x=3$ ?

Your turn

Question 1 of 2

For the displayed rule

x+3

when

x<-2

x^2

when

-2\le x\le2

, and

5-x

when

x>2

, which expression applies at

x=-3

Use all three rules

x+3

x^2

5-x

Explanation

Build it step by step

Difference Quotient

Worked example

Target

\frac{f(a+h)-f(a)}{h},\qquad h\ne0

Example: $f(x)=x^2+3x-1$

Build

f(a+h)

Compare

f(a)

Subtract

f(a+h)-f(a)

Factor

h(2a+h+3)

Cancel

2a+h+3

Difference quotient

f(x)=x^2+3x-1

, simplify

\frac{f(a+h)-f(a)}{h}

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Find f(a + h)

Which is the correct substitution?

Your turn

Question 1 of 2

In a difference quotient, what should be found before combining the numerator?

f(a+h)

and

f(a)

separatelyThe domain onlyThe range onlyA graph

Explanation

Make the decision

Domain · Mr. Azzam's Routine

Start with all real numbers. Then identify the structure and apply only the restriction it creates.

Interactive lab

Domain decision path

Identify the structure, state its restriction, and choose all permitted real inputs.

Structure: Fraction

f(x) = 1/(x − 2)

What is the domain?

FRACTION

\text{denominator}\ne0

Exclude denominator zeros.

EVEN ROOT

\text{radicand}\ge0

Keep values that make the root real.

EVEN ROOT AS THE DENOMINATOR

\text{radicand}>0

The root must exist and the denominator cannot be zero.

Worked example

Fractional expression

Find the domain of

f(x)=\frac{1}{x^2-5x+6}

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Find denominator zeros

What equation finds the excluded values?

Even root

Find the domain of

g(x)=\sqrt{9-x^2}

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Radicand restriction

Which inequality must be solved?

Root in denominator

Find the domain of

h(x)=\frac{1}{\sqrt{x-4}}

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Even-root condition

What does the square root require first?

Your turn

Question 1 of 2

Which restriction matches an even root in the numerator?

Radicand > 0Denominator = 0No restrictionRadicand ≥ 0

Key takeaway

Exam strategy

Common Mistakes → Better Habits

Common mistake

Repeated outputs

✕ Repeated output means not a function.

✓ Different inputs may share an output.

Common mistake

Function notation

✕ Drop parentheses for a negative input.

✓ Replace every x with the complete input.

Common mistake

Piecewise

✕ Pick the first rule in the list.

✓ Locate x → interval → rule → substitute.

Common mistake

Difference quotient

✕ Cancel before factoring.

✓ Subtract carefully, factor, then cancel.

Common mistake

Fraction domain

✕ Exclude numerator zeros.

✓ Exclude denominator zeros.

Common mistake

Even root as the denominator

✕ Use radicand ≥ 0.

✓ Use radicand > 0 because the denominator cannot be zero.

Key takeaway

Exam habit

Do not guess the method. Identify the structure first, then use the matching routine.

Your turn

Question 1 of 2

Two different inputs have the same output. Does that alone mean the relation is not a function?

Yes. Every input and output must be unique.No. Inputs do not matter.No. Repeated outputs are allowed.Yes. Outputs can never repeat.

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Function concept · input/output

Evaluating numerical inputs

Evaluating symbolic inputs

Piecewise functions & breakpoints

Difference quotient

Domain · denominator zeros

Domain · even roots

Domain · combined restrictions

Rate all skills to prepare your personalized practice path.

The Function Idea

1. Check the inputs

2. Conclusion

Your turn

Your turn · student questions

Connect the Representations

Your turn

Your turn · student questions

Domain & Range

1. Read the inputs

2. Read the outputs

Function Notation & Evaluating

1. Substitute carefully

2. Simplify

1. Replace x everywhere

2. Simplify

Your turn

Your turn · student questions

Piecewise Functions & Breakpoints

1. Left of 0

2. Exactly at 0

3. Between the breakpoints

1. Exactly at 3

2. Right of 3

Your turn

Your turn · student questions

Difference Quotient

1. Find f(a + h)

2. Find f(a)

3. Subtract carefully

4. Factor

5. Cancel h

Your turn

Your turn · student questions

Domain · Mr. Azzam's Routine

1. Find denominator zeros

2. Exclude them

1. Radicand restriction

2. Solve

1. Even-root condition

2. Denominator condition

3. Domain

Your turn

Your turn · student questions

Common Mistakes → Better Habits

Your turn

Your turn · student questions