Unit 5 · FunctionsF1 · Function Notation · Guidance

Function Notation & Composition

Treat functions as input-output rules and evaluate nested compositions inside out.

30 min
Unlocks: Flight Computer
Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

composition: using one function output as another function input

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

In $f(g(x))$ , complete the inside function $g$ first. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

f(g(x))

, complete the inside function

g

first. Keep exact values through the setup; round only when the stem asks.

Core relationship

(f\circ g)(x)=f(g(x))

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

For one or two evaluations, substitution is usually faster than graphing. Desmos function notation is helpful for checking nested compositions, provided the inside function is evaluated first.

Explore in Desmos

Evaluate a function at a negative input

Define the function, then type f(-3); for composition, enter the inside output next.

The calculator is live: change anything and see what happens.

Evaluate a function at a negative input

Degree mode

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Trap

The main trap

The exponent in

f(3)

belongs to the rule, not the function name.

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

f(x)=2x^2-1

; find

f(-3)

Replace every

x

-3

2(-3)^2-1=18-1

Which substitution preserves the negative input?

f(-3)=17

Why is the squared input positive 9?

Worked example

Example 2: ACT twist

Problem

f(x)=x-4

g(x)=3x+2

; find

g(f(5))

Inside first:

f(5)=1

Then

g(1)=3(1)+2

Which value becomes the input of $g$ ?

g(f(5))=5

Why must $f(5)$ be evaluated first?

Faded example · you finish it

Example 3: You finish it

Problem

h(x)=x^2+2

, find

h(h(1))

h(1)=3

Evaluate the outer function at the inner output 3.

Desmos way

Check $f(-3)$ in function notation

Define the function, then type f(-3); for composition, enter the inside output next.

Check f(-3) in function notation

Degree mode

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Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.

The decision that matters

Structure before arithmetic

Algebra or Desmos?

Evaluate a function at a negative input

The main trap

Check f(−3)f(-3)f(−3) in function notation

Check $f(-3)$ in function notation