Coordinates, Functions and Graphs, SAT math lesson

Press a button on a vending machine and one item drops out. Press the same button tomorrow and you get the same item. A function behaves like that machine, one input in and one output out, every time. The SAT shows functions as rules, as tables and as graphs, and many questions come down to moving a single input and output pair from one of these pictures to another.

See also Functions as Input-Output Relationships and Reading Graphs and Plotting Points in Foundations for Calculus and Function Notation and Evaluation in Precalculus.

Key terms

Ordered pairs and coordinates

Definition An ordered pair $(a, b)$ is a pair of numbers in which the order matters, so $(a, b) = (c, d)$ exactly when $a = c$ and $b = d$ . The coordinate plane has a horizontal $x$ -axis and a vertical $y$ -axis that cross at the origin $(0, 0)$ . The pair $(a, b)$ names the point reached by moving $\lvert a \rvert$ units to the right if $a > 0$ or to the left if $a < 0$ , and then $\lvert b \rvert$ units up if $b > 0$ or down if $b < 0$ . The number $a$ is the $x$ -coordinate of the point and $b$ is its $y$ -coordinate.

In plain words The first number says how far across, the second says how far up or down. Swap them and you usually get a different point.

Example The point $(3, -2)$ is $3$ units right of the origin and $2$ units down. The point $(-2, 3)$ is $2$ units left and $3$ units up, in a different part of the plane. Its $x$ -coordinate is $-2$ and its $y$ -coordinate is $3$ .

Function, input and output

Definition A function $f$ with domain $D$ assigns to each input $x$ in $D$ exactly one output, written $f(x)$ . Two different inputs may share an output, but a single input can never have two different outputs. When a function is given by a rule in $x$ , the input is the value substituted for $x$ and the output is the value the rule produces.

In plain words Each allowed input gets one answer, always the same one.

Example The pairs $(1, 4)$ , $(2, 4)$ and $(3, 9)$ describe a function. The inputs $1$ and $2$ share the output $4$ , and that is allowed. The pairs $(1, 4)$ and $(1, 5)$ cannot both belong to a function, because the input $1$ would have two outputs.

Function notation

Definition For a function $f$ and an input $a$ in its domain, $f(a)$ is the output of $f$ at $a$ . The parentheses here do not mean multiplication. The symbol $f(3)$ is a single number, the value of $f$ at $3$ , not $f$ times $3$ . To evaluate $f(a)$ from a rule, replace every $x$ in the rule by $a$ , written in parentheses.

In plain words Read $f(3)$ as “f of 3”, the output you get when $3$ goes in.

Example Let $f(x) = x^2 - 3x + 1$ . Then

\begin{aligned} f(-2) &= (-2)^2 - 3(-2) + 1 \\[0.4em] &= 4 + 6 + 1 = 11. \end{aligned}

The parentheses around $-2$ matter. Without them you might compute $-2^2 = -4$ and get the wrong output.

Table of values

Definition A table of values lists inputs together with their outputs, and each row is an ordered pair $(x, f(x))$ . A table gives information only about the inputs it lists. To find an output between or beyond the rows you need the rule or a statement such as “the function is linear”.

In plain words A table is a short list of input and output pairs. Each row is a point you could plot.

Example The table belongs to a linear function $g$ .

$x$	$0$	$1$	$2$	$3$
$g(x)$	$5$	$8$	$11$	$14$

The output grows by $3$ each time the input grows by $1$ , and $g(0) = 5$ , so the rule is $g(x) = 3x + 5$ . It matches every row, for instance $g(2) = 6 + 5 = 11$ . Because $g$ is linear, the rule also gives $g(10) = 35$ , an output the table never shows.

Graph of a function

Definition The graph of a function $f$ is the set of all points $(x, f(x))$ with $x$ in the domain of $f$ . A point $(a, b)$ lies on the graph exactly when $a$ is in the domain and $f(a) = b$ . Because each input has one output, a vertical line meets the graph of a function at most once. Before reading values from a graph, check the axis labels and how many units each grid square stands for.

In plain words Each point on the graph is one input and output pair. Its $x$ -coordinate is the input and its $y$ -coordinate is the output.

Example If the graph of $h$ passes through $(4, 7)$ , then $h(4) = 7$ . Now suppose the grid lines are $2$ units apart on both axes, and a point of the graph sits $3$ grid lines right of the origin and $5$ grid lines up. That point is $(6, 10)$ , not $(3, 5)$ , so $h(6) = 10$ .

Domain

Definition The domain of a function is the set of its allowed inputs. Unless a question says otherwise, the SAT takes the domain of a rule to be all real numbers for which the rule gives a real output. That excludes inputs that make a denominator zero and inputs that make the expression under a square root negative. In a real situation the context can restrict the domain further.

In plain words The domain is every number you are allowed to put in. Watch for division by zero, square roots of negatives, and limits set by the story.

Example For $f(x) = \sqrt{x - 2}$ the domain is all $x \ge 2$ , since $x - 2$ must not be negative. For $g(x) = \frac{5}{x + 4}$ it is every real number except $-4$ . A pool filled for $10$ minutes with $V(t) = 300 + 20t$ liters has domain $0 \le t \le 10$ , even though the formula itself works for any $t$ . Inside that domain an output has a meaning in the story. The value $V(4) = 380$ says the pool holds $380$ liters after $4$ minutes of filling.

Input expressions

Definition When the input is an expression, such as in $f(x + c)$ or $f(2x)$ , replace every $x$ in the rule by the whole expression in parentheses. In general $f(x + c)$ is not the same as $f(x) + c$ . The first changes the input before $f$ acts, the second changes the output after.

In plain words Whatever sits inside the parentheses is the input, even if it is a whole expression. Put all of it where $x$ was.

Example Let $f(x) = 3x - 4$ . Then

f(x + 2) = 3(x + 2) - 4 = 3x + 2,

while $f(x) + 2 = 3x - 2$ . At $x = 1$ the first is $f(3) = 5$ and the second is $f(1) + 2 = 1$ . With a table the idea is the same. If a table shows $f(5) = 11$ , then $f(x + 2)$ at $x = 3$ is $f(5)$ , which is $11$ .

Working back from an output

Definition To find the inputs that give an output $b$ , solve the equation $f(x) = b$ for all $x$ in the domain. On a graph these inputs are the $x$ -coordinates of the points where the horizontal line $y = b$ meets the graph. There may be exactly one such input, more than one, or none.

In plain words You know what came out and want to know what went in. Set the rule equal to the output and solve, or look along the horizontal line at that height.

Example For $f(x) = 3x - 4$ , the equation $f(x) = 11$ gives $3x = 15$ , so $x = 5$ . For $p(x) = x^2$ , the equation $p(x) = 9$ has two solutions, $x = 3$ and $x = -3$ , and $p(x) = -1$ has none. In the Desmos graphing calculator, graph $y = x^2$ and $y = 9$ and click each point where they meet. The coordinates shown are $(-3, 9)$ and $(3, 9)$ , and the inputs are their $x$ -coordinates.

Check yourself

The function $f$ is defined by $f(x) = x^2 - 2x$ . What is the value of $f(-3)$ ? (A) $-15$ (B) $-3$ (C) $3$ (D) $15$

Show the answer

The correct choice is (D). Substituting with parentheses gives $f(-3) = (-3)^2 - 2(-3) = 9 + 6 = 15$ . Choice (C) comes from writing $9 - 6$ and losing the sign of $-2 \cdot (-3)$ . Choice (B) comes from computing $-3^2$ as $-9$ , and (A) makes both mistakes.

The function $h$ is defined by $h(x) = 2x - 5$ . If $h(a + 3) = 17$ , what is the value of $a$ ?

Show the answer

The value is $8$ . The input is $a + 3$ , so $h(a + 3) = 2(a + 3) - 5 = 2a + 1$ . Setting $2a + 1 = 17$ gives $a = 8$ . Check that $h(11) = 22 - 5 = 17$ .

The table shows some values of the linear function $f$ . Which equation defines $f$ ? (A) $f(x) = 2x + 4$ (B) $f(x) = -2x + 4$ (C) $f(x) = -2x - 4$ (D) $f(x) = 4x - 2$

$x$	$-1$	$0$	$2$	$5$
$f(x)$	$6$	$4$	$0$	$-6$

Show the answer

The correct choice is (B). The row for $x = 0$ shows $f(0) = 4$ , which rules out (C) and (D). Choice (A) gives $f(-1) = 2$ , not $6$ . Choice (B) matches every row, for example $f(5) = -10 + 4 = -6$ .

Coordinates, Functions and Graphs

Theory, part 19

Practice, part 20

Key terms

Ordered pairs and coordinates

Function, input and output

Function notation

Table of values

Graph of a function

Domain

Input expressions

Working back from an output

Check yourself

How this lesson connects

Builds on

Used later in