Functions

A1.E.1Function notation & evaluation01/05

Function notation and evaluation.

Step01of 05

A function is a rule that takes an input and returns exactly one output. $f (x)$ reads "f of x" and means "the output of the function $f$ when the input is x".

f (x) = 2 x + 3

Read it as a recipe: take an input, double it, add 3. $f (x)$ is NOT multiplication of $f$ and x — the parentheses mean "evaluated at."

A1.E.1 — Function notation and evaluation.

Step 01 of 05

A function is a rule that takes an input and returns exactly one output. $f (x)$ reads "f of x" and means "the output of the function $f$ when the input is x".

f (x) = 2 x + 3

Read it as a recipe: take an input, double it, add 3. $f (x)$ is NOT multiplication of $f$ and x — the parentheses mean "evaluated at."

Step 02 of 05

Evaluating means: substitute the input wherever you see x.

f (5) = 2 (5) + 3 = 13

f (0) = 2 (0) + 3 = 3

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

Always wrap the substituted value in parentheses — same habit as evaluating any algebraic expression.

Step 03 of 05

The input can be an expression, not just a number. Wherever x appears, drop in the entire input.

f (x + 4) = 2 (x + 4) + 3 = 2 x + 8 + 3 = 2 x + 11

f (3 x) = 2 (3 x) + 3 = 6 x + 3

The substitution rule doesn't change. Treat the input as a single indivisible chunk.

Step 04 of 05

One input → one output. That's the rule that makes something a function. If a single input could give two different outputs, it isn't a function.

$y = x^{2}$	→function — every x gives one y
$y^{2} = x$	→NOT a function — $x = 4$ gives $y = \pm 2$
"vertical line test"	→any vertical line hits the graph at most once → function

Step 05 of 05

Function names beyond $f$ . Any letter works — $g (x)$ , $h (t)$ , $P (n)$ . The letter inside the parens is the input variable; the letter outside is the function name.

P (n) = 50 n + 200 ⟹ P (10) = 700

Key insight

Function notation is a compact way to say "for ANY input, here's the output." It looks fancy, but it's just substitution with the input wrapped in parens.

A1.E.2 — Domain and range.

Step 01 of 05

Domain = all valid inputs (the x-values you're allowed to plug in). Range = all outputs that result (the y-values that come out).

For most "nice" linear functions, both are all real numbers. The skill is recognizing the cases where they aren't.

Step 02 of 05

Case 1 — square roots. The expression under the radical can't be negative.

f (x) = x - 3

Need $x - 3 \geq 0$ , so $x \geq 3$ . Domain: $[3, \infty)$ . Range: $[0, \infty)$ (square roots are never negative).

Step 03 of 05

Case 2 — fractions. The denominator can't be zero.

f (x) = \frac{1}{x - 5}

Need $x - 5 \neq = 0$ , so $x \neq = 5$ . Domain: all real numbers except $5$ . Range: all real numbers except $0$ (the function never outputs zero — a fraction with a nonzero numerator can't equal zero).

Step 04 of 05

Case 3 — quadratics and parabolas. Domain is always all real numbers; range depends on the vertex.

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

Vertex at $(0, - 4)$ , opens upward. Range: $[- 4, \infty)$ . The vertex's y-value is the lowest (or highest, if it opens down) the function ever reaches.

Step 05 of 05

Reading domain/range from a graph.

Domain	→shadow the graph onto the x-axis — that span IS the domain
Range	→shadow the graph onto the y-axis — that span IS the range
Open dot	→endpoint NOT included — use parens $($ or $)$
Closed dot	→endpoint IS included — use brackets $[$ or $]$

Key insight

Three flags trigger restrictions: square roots (need

\geq 0

inside), fractions (denominator

\neq = 0

), and parabolas (range capped at vertex). Linear functions have no restrictions — both domain and range are all reals.

A1.E.3 — Average rate of change.

The function

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

Step 01 of 05

For a linear function, slope is constant — the same everywhere. For nonlinear functions, "slope" changes from point to point. The average rate of change over an interval is the slope of the secant line connecting the endpoints.

ARC = \frac{f ( x _{2} ) - f ( x _{1} )}{x _{2} - x _{1}}

Same formula as slope. The only twist: the y-values come from evaluating the function at the endpoints.

Step 02 of 05

Find the average rate of change of $f (x) = x^{2} + 1$ on $[1, 4]$ . Step 1 — evaluate the endpoints.

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

f (4) = 4^{2} + 1 = 17

Step 03 of 05

Step 2 — apply the slope formula.

ARC = \frac{f ( 4 ) - f ( 1 )}{4 - 1} = \frac{17 - 2}{3} = \frac{15}{3} = 5

On $[1, 4]$ , the function rises by an average of 5 units per unit of x. The actual graph isn't a straight line, but if you drew a chord from $(1, 2)$ to $(4, 17)$ , that chord has slope 5.

Step 04 of 05

Different intervals give different rates — that's the whole point for a nonlinear function. Same $f$ , on $[0, 2]$ :

ARC = \frac{f ( 2 ) - f ( 0 )}{2 - 0} = \frac{5 - 1}{2} = 2

Average rate $2$ on the lower interval, $5$ on the upper. The parabola is steepening — that's exactly what changing rates tells you.

Step 05 of 05

Real-world reading. If $f (t)$ is a position at time $t$ , then ARC over $[t_{1}, t_{2}]$ is the average velocity. If $f (t)$ is profit at month $t$ , ARC is the average monthly profit growth. Same formula, lots of contexts.

Key insight

Average rate of change = slope of the secant. For lines it's the slope. For curves it's the slope of the chord between the endpoints — the value changes when you change the interval.

A1.E.4 — Piecewise and step functions.

The piecewise function

f (x) = ⎩ ⎨ ⎧ - x + 1, x^{2}, 5, x < 0 0 \leq x \leq 2 x > 2

Step 01 of 05

A piecewise function uses different rules on different parts of the domain. To evaluate, look at WHICH piece your input falls into, then apply that piece's rule.

Step 02 of 05

Evaluate $f (- 3)$ . The input $- 3$ is $< 0$ , so use the FIRST rule.

f (- 3) = - (- 3) + 1 = 3 + 1 = 4

Don't accidentally plug into $x^{2}$ just because it's there — that piece only applies on $[0, 2]$ .

Step 03 of 05

Evaluate $f (1)$ . The input $1$ satisfies $0 \leq x \leq 2$ , so use the SECOND rule.

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

Evaluate $f (7)$ . Input $7 > 2$ — third rule.

f (7) = 5

The third piece is a constant piece — output is $5$ regardless of input.

Step 04 of 05

Boundaries get careful treatment. What's $f (0)$ ? Both the first and second pieces involve $x = 0$ , but the first uses strict $<$ , so $0$ falls under the second piece:

f (0) = 0^{2} = 0

The choice of $<$ vs. $\leq$ at each boundary is deliberate — exactly one piece "owns" each input.

Step 05 of 05

Step functions are a special piecewise: each piece is a constant. The classic example is shipping cost.

Cost (w) = ⎩ ⎨ ⎧ 5, 8, 12, 0 < w \leq 1 1 < w \leq 3 3 < w \leq 5

Graph looks like flat segments separated by vertical jumps — a staircase. Endpoint conventions (open vs. closed dots) match the $<$ / $\leq$ choices in the definition.

Key insight

For piecewise: pick the piece whose condition the input satisfies, then evaluate normally. The "magic" is just routing — the math inside each piece is the same algebra you already know.

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A1.E.1 – A1.E.4 · CCSS F.IF.A.2 / F.IF.A.1 / F.IF.B.6 / F.IF.C.7b·Full curriculum →