ACT Functions: Complete Guide to Notation, Graphs & Transformations

Functions are a major part of current ACT Math.

ACT places Functions inside the Preparing for Higher Math category, and ACT says Functions account for roughly 17–20% of the Math section.

That makes function fluency one of the highest-value ACT Math skills. For the full rundown of the current exam, see our Enhanced ACT guide, and browse the rest of our ACT guides for other sections.

What is a function?

A function takes an input and produces exactly one output.

If:

[ f(x)=3x+2 ]

then:

[ f(4)=3(4)+2=14 ]

The notation (f(4)) does not mean (f\times4).

It means:

put 4 into the rule.

Evaluating a function

Given:

[ g(x)=x^2-5x+6 ]

Find (g(3)):

[ g(3)=9-15+6=0 ]

Solving from an output

If:

[ f(x)=2x+7 ]

and:

[ f(x)=19 ]

then:

[ 2x+7=19 ]

[ x=6 ]

Functions from tables

The output rises by 3 each time x rises by 1.

That suggests:

[ f(x)=3x+2 ]

Reading a function graph

For the graph of (y=f(x)):

• (f(2)) means the y-value when x=2
• (f(x)=5) asks for x-values where the graph has y=5
• (f(x)=0) means the x-intercepts

Those are three different questions.

Reading graphs carefully is also central to ACT Science, where the data and graphs guide covers the same skill applied to experiments and tables.

Transformations

Starting from (y=f(x)):

Up

[ f(x)+4 ]

moves the graph 4 units up.

Down

[ f(x)-4 ]

moves it 4 units down.

Right

[ f(x-4) ]

moves it 4 units right.

Left

[ f(x+4) ]

moves it 4 units left.

Horizontal transformations feel reversed because the change occurs inside the input.

Domain

The domain is the set of allowed inputs.

Example:

[ f(x)=\frac{1}{x-3} ]

x cannot equal 3.

For:

[ f(x)=\sqrt{x-5} ]

in real numbers:

[ x\ge5 ]

Function composition

Suppose:

[ f(x)=x+2 ]

[ g(x)=3x ]

Find:

[ g(f(4)) ]

First:

[ f(4)=6 ]

Then:

[ g(6)=18 ]

Work from the inside outward.

Linear vs nonlinear functions

Linear

[ f(x)=mx+b ]

Constant rate of change.

Quadratic

[ f(x)=ax^2+bx+c ]

Rate of change is not constant.

Exponential

[ f(x)=ab^x ]

Changes by a constant factor over equal intervals.

ACT questions often ask you to recognize the model from a table, graph or context.

Original practice

1

If (f(x)=5x-4), what is (f(6))?

2

If (g(x)=x^2+3) and (g(a)=28), what are the possible values of a?

3

Which expression shifts (y=f(x)) 7 units right?

A. (f(x)+7)
B. (f(x)-7)
C. (f(x-7))
D. (f(x+7))

4

If (f(x)=2x+1) and (g(x)=x^2), what is (g(f(2)))?

5

For (h(x)=1/(x+9)), which value is excluded from the domain?

Answers

1. 26
2. 5 and -5
3. C
4. 25
5. -9

Common ACT traps

• treating function notation as multiplication;
• reversing horizontal shifts;
• confusing x-values with y-values;
• forgetting domain restrictions;
• doing composition in the wrong order.

Once these traps feel manageable, check upcoming ACT test dates to plan when to put this practice to the test.

Official source

• ACT Math Test Description: https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/description-of-math-test.html

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