Digital SAT · Math · Unit 2 · Advanced Math

Lesson 2.3 · Functions & Function Notation

On the test: 3–5 questions every test — the most common Advanced Math topic

Big idea. A function is a machine: one input goes in, exactly one output comes out. f(x) is the name of the output, not a multiplication. Every question is one of two moves: put a number IN (evaluate) or know the output and find what went in (solve).

🧑‍🏫 Rule 1 · Evaluate: replace EVERY x with the inputstep 1 of 4

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📋 The question · SAT · December 2023 · Math 800 Book Q15
$g (x) = 6 x + 7$ . What is $g (3)$ ?
1. A) $\frac{5}{6}$
2. B) $\frac{25}{6}$
3. C)5
4. D)25
the question
$g (x) = 6 x + 7, g (3)$
December 2023. Write the rule again with 3 in brackets wherever $x$ was.

🧑‍🏫 Rule 2 · Solve: the output is given, find the inputstep 1 of 4

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📋 The question · SAT · June 2023 · Math 800 Book Q5
$f (x) = 6 (2 x + 4)$ . For what value of $x$ does $f (x) = 48$ ?
1. A)2
2. B)6
3. C)8
4. D)22
the question
$f (x) = 6 (2 x + 4) = 48$
June 2023: “for what $x$ does $f (x) = 48$ ?” — set the RULE equal to 48.

🧑‍🏫 Rule 3 · Outputs combinedstep 1 of 4

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📋 The question · SAT · May 2024 · Math 800 Book Q32
If $f (x) = x + 2$ and $g (x) = 2 x$ , what is $3 f (- 5) - g (- 5)$ ?
1. A) $- 19$
2. B) $- 3$
3. C)1
4. D)7
the question
$3 f (- 5) - g (- 5), f = x + 2, g = 2 x$
May 2024: run each machine on $- 5$ first.

Desmos move · solve by graphing

Type y = 6(2x+4) and y = 48; tap the crossing. For roots and absolute values this avoids sign slips.

🧑‍🏫 Rule 4 · f(0) is where the graph meets the y-axisstep 1 of 3

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why
$x = 0 on the y -axis$
Every point on the $y$ -axis has $x = 0$ , so its height is $f (0)$ .

🧑‍🏫 Rule 5 · Average rate of change = slope between two pointsstep 1 of 4

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the formula
$\frac{f ( b ) - f ( a )}{b - a}$
Even on a curve, “average rate” is the straight-line slope between the two points.

Real question 1SAT · August US 2024 · Math 800 Book Q38type the answer

$figure from the test$

Figure from the test paper — our labels appear after you answer.

The graph shows the momentum $y$ , in newton-seconds, of an object $x$ seconds after it started moving, for $0 \leq x \leq 8$ . What is the average rate of change, in newton-seconds per second, from $x = 2$ to $x = 6$ ?

🧑‍🏫 Rule 6 · Shift up: add to every outputstep 1 of 3

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📋 The question · SAT · May 2024 · Math 800 Book Q29
$f (x) = (x - 9) (x - 6) (x - 5)$ . The graph of $y = g (x)$ is $y = f (x)$ translated up 4. What is $g (0)$ ?
(type the answer)
the question
$f (x) = (x - 9) (x - 6) (x - 5), g = f + 4$
May 2024: $g (0)$ ?

🧑‍🏫 Rule 7 · Undo a built function to recover fstep 1 of 4

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📋 The question · SAT · June 2024 · Math 800 Book Q36
A table: $g (- 20) = 4$ , $g (- 8) = 0$ , $g (10) = 6$ , where $g (x) = \frac{f ( x )}{x + 5}$ and $f$ is linear. What is the $y$ -intercept of $y = f (x)$ ?
1. A) $(0, - 8)$
2. B) $(0, 5)$
3. C) $(0, 8)$
4. D) $(0, 40)$
the question
$g (x) = \frac{f ( x )}{x + 5}$
June 2024: a table of $g$ ; $f$ is linear. Find its $y$ -intercept.

🧑‍🏫 Rule 8 · A parabola is symmetric about its vertexstep 1 of 4

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📋 The question · SAT · June 2023 · Math 800 Book Q1
The graph of the quadratic function $y = f (x)$ intersects the $x$ -axis when $x = 39$ and when $x = p$ . The maximum value of $f$ occurs at $(14, m)$ . What is $p$ ?
(type the answer)
the question
$zeros 39, p; max at x = 14$
June 2023.

🎛 Move it · zeros and the shape

Every zero is an $x$ -intercept; count them in the formula (repeated factors count once).

−

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

zero 1-3zero 21zero 34

Each zero r gives a factor (x − r) and an x-intercept (r, 0). Sliding a zero slides that crossing — nothing else.

f(0) = a × (−r₁)(−r₂)… = 12. The sign flips at every zero.

🧑‍🏫 Rule 9 · Same AMOUNT added → linear; same FACTOR multiplied → exponentialstep 1 of 3

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📋 The question · SAT · March 2024 · Math 800 Book Q25
For $x > 0$ , $f (x)$ equals 111% of $x$ . Which could describe $f$ ?
1. A)Decreasing exponential
2. B)Decreasing linear
3. C)Increasing exponential
4. D)Increasing linear
linear
$f (x) = 111% of x = 1.11 x$
March 2024: a constant times $x$ — a straight line through the origin, increasing (option D).

🎛 Move it · absolute value = distance

May 2024: $7∣ x - 7∣ = 42$ → $∣ x - 7∣ = 6$ . Two answers symmetric about 7 add to 14.

|x − 6| = 13

-30-20-100102030h = 6

centre h = 6distance k = 13

Two answers: 6 − 13 = -7 and 6 + 13 = 19. Their sum is 12 = 2 × 6 — always twice the centre.

🧑‍🏫 Rule 10 · Counting x-interceptsstep 1 of 4

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📋 The question · SAT · November 2023 · Math 800 Book Q10
How many $x$ -intercepts does $f (x) = x (x - 8)^{2} (x - 13)^{3}$ have?
1. A)0
2. B)1
3. C)2
4. D)3
the question
$f (x) = x (x - 8)^{2} (x - 13)^{3}$
November 2023: how many $x$ -intercepts?

🧑‍🏫 Rule · Composition: inside firststep 1 of 3

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📋 The question · Gen Z Study
If $f (x) = 2 x + 1$ and $g (x) = x^{2} - 3$ , what is $f (g (2))$ ?
(type the answer)
inside
$g (2) = 4 - 3 = 1$
Work from the inside bracket out.

🧑‍🏫 Rule · Domain: inputs that make sensestep 1 of 3

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no division by zero
$\frac{1}{x - 4} \Rightarrow x \neq = 4$

Take with you

▸ $f (a)$ : replace every $x$ with $a$ . Solving: set the rule equal to the output.
▸ $f (0)$ is the $y$ -intercept; a point $(a, b)$ means $f (a) = b$ .
▸Average rate of change $= \frac{f ( b ) - f ( a )}{b - a}$ ; “times as much” is a ratio.
▸ $f (x) + k$ adds $k$ to every output; undo built functions by multiplying back.
▸A parabola’s vertex is halfway between equal outputs.
▸Constant difference → linear; constant ratio → exponential.

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