Function notation — f(x), g(x): Question bank

Before we start, three shared pictures to hold in your head.

The two trap pictures you will need — composition order and the shift transformations — are shown here so the geometry is concrete before you test yourself.

True or false — justify

True or false: If

f (x) = 2 x + 1

then

f = 2 x + 1

False.

f

is the rule (a machine);

2 x + 1

is a number-valued expression. You may only write

f (x) = 2 x + 1

, equating the output to the expression, never the machine itself to a number.

True or false:

f (3)

is a function.

False.

f

is the function;

f (3)

is a single number — the output of that function at the one input

3

True or false: The letters

f

and

x

carry meaning, so I must always use exactly those.

False. They are just names.

g (t) = 2 t + 1

defines the same machine as

f (x) = 2 x + 1

; only the rule matters, not the letters chosen for name or input.

True or false:

f (x + 2) = f (x) + 2

for the rule

f (x) = x^{2}

False.

f (x + 2) = (x + 2)^{2} = x^{2} + 4 x + 4

, but

f (x) + 2 = x^{2} + 2

. The first changes the input before squaring; the second changes the output after.

True or false:

f (2 x)

always equals

2 f (x)

False in general. For

f (x) = x^{2}

f (2 x) = 4 x^{2}

but

2 f (x) = 2 x^{2}

. Only special ("linear through origin") rules like

f (x) = 5 x

let the multiplier slide out.

True or false:

f (g (x))

and

g (f (x))

describe the same combined machine.

False. Composition is not commutative —

f (g (x))

does

g

first then

f

, while

g (f (x))

does

f

first then

g

; different order, generally different output (see figure s02).

True or false:

y = x^{2}

and

f (x) = x^{2}

describe the same object.

True. They are the same relationship seen two ways: the equation stresses the

x

–

y

link (with

y

the output height), the notation stresses the named rule. Write

y = f (x)

to bridge them.

True or false: If

f (a) = f (b)

then

a

must equal

b

False. Different inputs can share an output — e.g.

f (x) = x^{2}

gives

f (3) = f (- 3) = 9

. (Only one-to-one functions forbid this, which is why they alone have inverses; see Inverse functions.)

True or false:

f^{- 1} (x)

means

\frac{1}{f ( x )}

False.

f^{- 1}

is the inverse machine that undoes

f

, not a reciprocal. The

- 1

is a "reverse the process" mark, not an exponent. (See Inverse functions.)

True or false: The name inside

f (\cdot)

tells you what to compute.

False. The name

f

tells you which rule; the thing in the slot (the dot) is the input fed to that rule. Same input, different name → different answer.

Why questions

Why do we bother naming the rule

f

instead of always writing

y = \dots

So we can name several rules at once (

f, g, h

), combine them like

f (g (x))

, and talk about a function's properties without picking specific numbers.

Why must we substitute the entire input expression, brackets and all, into the rule?

Because the machine acts on whatever single quantity you hand it; treating

(x + 2)

as one input keeps the substitution unambiguous and prevents dropping cross-terms like

4 x

(x + 2)^{2}

Why is reading composition right-to-left the correct habit?

Because in

f (g (x))

the input

x

physically enters

g

first (the innermost bracket) and only the result travels outward to

f

; the layout mirrors the flow of the data (figure s02).

Why can't we write "

f = x^{2} + 1

It equates a process (the machine) to a number-valued expression. A process has no numeric value until fed an input, so the equation is a category error; write

f (x) = x^{2} + 1

instead.

Why does "

f (x)

" being a number, not a rule, actually matter in practice?

Because once

f (x)

is a number, you can drop it into another machine — that is exactly what makes composition, e.g.

g (f (x))

, legal and unambiguous (see Composite functions).

Why does swapping the input letter (e.g.

f (t)

instead of

f (x)

) not change the function?

The input letter is a placeholder — the empty slot

f (\cdot)

filled with any letter; the rule pattern ("square it, add one") is what defines the machine, so any consistent letter names the same function.

Edge cases

What is

f (0)

for

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

, and what does it mean geometrically?

f (0) = 5

; every term with

x

vanishes, leaving the constant, which is the $y$ -intercept — where the curve crosses the vertical (

y

) axis (see Graphing functions).

f

is a constant function

f (x) = 7

, what is

f (f (x))

Still

7

. Feeding

7

back into a machine that always returns

7

changes nothing — constant functions absorb any input.

Can the input to

f

itself be another function's output that is undefined?

If the inner output is undefined (input outside its domain), the whole composition is undefined — you cannot feed a machine something that never came out. Domains must line up (see Composite functions).

For

f (x) = x^{2}

, is

f (f (- 3))

different from

f (- 3)

Yes.

f (- 3) = 9

, then

f (9) = 81

, so nesting applies the rule twice — very different from applying it once.

f (x) = x

(the identity machine) a "real" function, and what does it do in composition?

Yes; it outputs its input unchanged, so

f (g (x)) = g (x)

and

g (f (x)) = g (x)

— it acts like the number

1

does for multiplication, leaving other machines untouched.

What happens to

f (x + 2)

versus

f (x) + 2

as a graph transformation?

f (x + 2)

shifts the graph left by 2 (input changed first);

f (x) + 2

shifts it up by 2 (output changed after). Same

2

, perpendicular effects — compare the two shifted curves in figure s03 (see Function transformations).

Recall One-line summary to carry away

$f$ is the machine, $x$ is the input, $f (x)$ is the output number — mix these up and every trap above springs shut.

Connections

Parent — the notation this bank stress-tests
Composite functions — order and domain-matching traps (figure s02)
Inverse functions — the $f^{- 1}$ -is-not-reciprocal trap
Function definition and domain — domain, codomain and undefined-input edge cases (figure s01)
Graphing functions · Function transformations — geometric meaning of $f (0)$ , and $f (x + 2)$ vs $f (x) + 2$ (figure s03)

Question bank — Function notation — f(x), g(x)

True or false — justify

Spot the error

Why questions

Edge cases

Connections