8.2 Domain, Range, and Interpreting Functions in Context

The Largest AAF Content Area

Functions: Creating functions using function notation, evaluating linear and quadratic functions, graphing functions, and interpreting functions within a context.

At 2 to 4 of the 20 questions, Functions is tied for the largest allocation on the AAF test. The final clause — interpreting within a context — is the part students prepare for least.

What Makes a Relation a Function

A function assigns exactly one output to each input. One $x$, one $y$.

The vertical line test: if any vertical line crosses a graph more than once, the graph is not a function.

• A parabola opening upward is a function.
• A circle is not — most vertical lines hit it twice.
• A sideways parabola $x = y^2$ is not.

In a table, a repeated input with different outputs disqualifies the relation:

Not a function: $x = 2$ maps to both 5 and 7. Repeated outputs are fine — $y$ may repeat for different $x$.

Domain and Range From an Equation

Domain = permitted inputs. Range = resulting outputs.

Start from "all real numbers" and remove what breaks:

$f(x) = \dfrac{x+1}{x-7}$ → domain: all reals except $x = 7$ $g(x) = \sqrt{x+5}$ → $x + 5 \ge 0$ → domain: $x \ge -5$ $h(x) = \log(x - 2)$ → $x - 2 > 0$ → domain: $x > 2$

Note the logarithm uses a strict inequality — $\log 0$ is undefined, not zero.

Range

Range usually requires thinking about the function's shape.

$f(x) = x^2$ → range $y \ge 0$ $f(x) = x^2 + 3$ → range $y \ge 3$ $f(x) = -x^2 + 5$ → opens downward, maximum 5 → range $y \le 5$ $f(x) = \sqrt{x}$ → range $y \ge 0$

For a quadratic, the range boundary is the vertex $y$-value: minimum if the parabola opens up ($a > 0$), maximum if it opens down ($a < 0$).

$f(x) = 2x^2 - 8x + 1$. Vertex at $x = -\dfrac{b}{2a} = \dfrac{8}{4} = 2$; $f(2) = 8 - 16 + 1 = -7$. Since $a = 2 > 0$, range is $y \ge -7$.

Domain and Range From a Graph

• Domain: the horizontal extent — how far left and right the graph goes.
• Range: the vertical extent — how far down and up.

Read endpoints carefully: a closed dot includes the value, an open dot excludes it, and an arrow indicates the graph continues indefinitely.

Interpreting Functions in Context

This is the distinguishing AAF skill, and the answers are sentences, not numbers.

A company models profit as $P(x) = -2x^2 + 240x - 3000$, where $x$ is units produced.

What does $P(50)$ represent?

The profit, in dollars, from producing 50 units. Computing: $-2(2500) + 12000 - 3000 = -5000 + 9000 = 4000$, so producing 50 units yields $4,000 profit.

What does the vertex represent?

$x = -\dfrac{240}{2(-2)} = 60$. Because $a < 0$ the parabola opens downward, so the vertex is a maximum: profit peaks at 60 units. $P(60) = -7200 + 14400 - 3000 = 4200$, a maximum profit of $4,200.

What do the $x$-intercepts represent?

Break-even points, where profit is zero.

What is a reasonable domain?

$x \ge 0$ — you cannot produce a negative number of units. This is a contextual restriction, not an algebraic one: the formula accepts negatives happily, but the situation does not.

Contextual Domain Restrictions

This distinction recurs across applied items:

A ball's height is $h(t) = -16t^2 + 48t + 6$ feet after $t$ seconds.

Algebraically the domain is all reals. Contextually it runs from $t = 0$ until the ball lands ($h = 0$), because negative time is meaningless and the model stops applying after impact.

Reading Function Behavior

Applied questions frequently ask about behavior rather than values:

• Increasing: graph rises left to right — the quantity is growing.
• Decreasing: graph falls — the quantity is shrinking.
• Maximum/minimum: the vertex or a turning point — peak or trough.
• Zeros / $x$-intercepts: where output is zero — break-even, landing, depletion.
• $y$-intercept: the value at input zero — initial or starting amount.

For $h(t) = -16t^2 + 48t + 6$: the $y$-intercept 6 is the release height, the vertex is the maximum height, and the positive zero is when the ball lands.

Answering these correctly requires naming the quantity and its units — "the maximum height in feet," not "the vertex."