ACT · Domain 3

Functions

12–15% of the ACT math section

Notation, graphs and transformations, across every family the test uses: linear, quadratic, exponential, logarithmic and piecewise. The ACT tests logarithms and composition, which the SAT leaves alone.

Functions · Topic 1 of 3

Notation, Graphs & Transformations

$f(x)$ is an instruction: take $x$ , do this to it. $f(g(x))$ is two instructions, and the inside one runs first. Most composition mistakes are order mistakes, and the ACT puts the other order among the choices.

The problem

If

f(x) = 2x + 1

and

g(x) = x^2

, what is

f(g(3))

?

A $19$
B $49$
C $7$
D $37$

Working1 / 3

Inside first $g(3) = 3^2 = 9$

Functions · Topic 2 of 3

Linear, Quadratic, Exponential & Logarithmic

A logarithm is an exponent written the other way round: $\log_b x = y$ means $b^y = x$ . That one sentence answers most ACT log questions, because they are asking you to rewrite and read off, not to compute.

The problem

If

\log_3 x = 4

, what is

x

?

F $81$
G $12$
H $64$
J $7$

Working1 / 2

Rewrite the log as an exponent $3^4 = x$

Functions · Topic 3 of 3

Arithmetic & Geometric Sequences

An arithmetic sequence adds the same number each step; a geometric one multiplies by the same number. Each has a formula for the $n$ th term, and the ACT asks for a far term — the 40th — or the sum of the first $n$ . Write the first term and the step or the ratio down before anything else, and count the steps: the 5th term is four steps from the first.

The problem

The first term of a geometric sequence is 5 and the common ratio is 3. What is the 5th term of the sequence?

A $405$
B $1{,}215$
C $135$
D $17$

Working1 / 3

The $n$ th term of a geometric sequence $a_n = a_1 \cdot r^{\,n - 1}$