What Is a Function?: Practice

A function pairs each allowed input with a single, definite output. Two different inputs may share an output, but one input can never produce two different outputs.

$\text{each input} \;\longrightarrow\; \text{exactly one output}$

Only the choice naming exactly one output for each input captures the definition.

Substitute $3$ for $x$ and simplify.

$f(3) = 3 + 4 = 7$

The input is $3$ and the output is $7$.

The rule doubles the input, so replace $x$ with $5$.

$f(5) = 2 \cdot 5 = 10$

The input is the value inside the parentheses; the output is what the function equals.

$f(\underbrace{3}_{\text{input}}) = \underbrace{7}_{\text{output}}$

So the input is $3$.

Squaring means multiply the input by itself, so replace $x$ with $4$.

$g(4) = 4^2 = 16$

The distractor $8$ comes from doubling instead of squaring.

Substitute $2$ for $x$, multiplying before subtracting.

$f(2) = 5 \cdot 2 - 2 = 10 - 2 = 8$

Substitute $4$ for $x$, multiplying before adding.

$f(4) = 2 \cdot 4 + 1 = 8 + 1 = 9$

The definition forbids one input from having two outputs, but it says nothing against two inputs sharing an output. Squaring is an example.

$3 \mapsto 9 \quad \text{and} \quad (-3) \mapsto 9$

So the statement is true.

Read the output listed for the input $2$.

$f(2) = 7$

The input $2$ is paired with the output $7$.

Square the input first, then subtract $1$.

$f(3) = 3^2 - 1 = 9 - 1 = 8$

The name of the function is the letter written just before the parentheses.

$\underbrace{f}_{\text{name}}(x) = 2x + 1$

So $f$ names the rule, while $x$ is the input.

Substitute $0$ for $x$.

$f(0) = 4 \cdot 0 = 0$