Functions on the Digital SAT: Notation, Graphs, Transformations & Practice

Functions appear across SAT Math, especially in Advanced Math and Algebra.

The key idea:

A function is a rule that assigns each allowed input exactly one output.

If:

[ f(x)=2x+5 ]

then (f(3)) means:

[ 2(3)+5=11 ]

It does not mean (f\times3).

Function notation

Given:

[ g(x)=x^2-4x+1 ]

Find (g(5)):

[ 25-20+1=6 ]

Solve from a function value

If:

[ f(x)=3x-7 ]

and (f(x)=20):

[ 3x-7=20 ]

[ x=9 ]

Tables

A function can be represented by a table.

The output increases by 3 for every +1 in x.

Linear rule:

[ f(x)=3x+2 ]

Graph interpretation

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

• (f(2)) = y-value when x=2
• solution to (f(x)=5) = x-value(s) where graph crosses y=5
• x-intercepts = solutions to (f(x)=0)

Transformations

Starting from (y=f(x)):

Vertical

[ f(x)+k ]

shifts up (k).

[ f(x)-k ]

shifts down (k).

Horizontal

[ f(x-h) ]

shifts right (h).

[ f(x+h) ]

shifts left (h).

Horizontal signs feel reversed because the transformation occurs inside the input.

Linear functions

[ f(x)=mx+b ]

• m = rate of change / slope
• b = initial value / y-intercept

In context, always interpret units.

If:

[ C(t)=18+4.5t ]

where C is dollars and t is rentals:

18 = fixed fee
4.5 = dollars per rental.

These slope-and-intercept ideas are the same ones tested directly in linear equations questions.

Exponential functions

[ f(x)=ab^x ]

• a = initial value at x=0
• b = growth/decay factor

If b>1: growth
If 0<b<1: decay

Quadratic functions

[ f(x)=a(x-h)^2+k ]

Vertex:

[ (h,k) ]

This makes minimum/maximum questions fast. Finding that vertex ties directly into the vertex-form work covered in quadratic equations.

Composition

If:

[ f(x)=2x+1 ]

and

[ g(x)=x^2 ]

then:

[ g(f(3)) ]

First:

[ f(3)=7 ]

Then:

[ g(7)=49 ]

Work inside-out.

Domain

Domain = allowed inputs.

Example:

[ f(x)=\frac{1}{x-4} ]

x cannot equal 4. For rational functions with a more complex denominator, factoring first — as in polynomials and factoring — makes the excluded values easier to spot.

For:

[ f(x)=\sqrt{x-2} ]

in real numbers:

[ x\ge2 ]

Desmos strategy

Desmos is excellent for:

• evaluating graphs;
• intersections;
• zeros;
• vertex;
• transformations;
• comparing function models.

But if the function is:

[ f(x)=3x+2 ]

and you need (f(4)), substitution is faster.

Original mini-practice

1

If (f(x)=4x-3), what is (f(6))?

2

If (g(x)=x^2+2) and (g(a)=27), what are the possible values of a?

3

The graph of (y=f(x)) is shifted 5 units right. Which expression represents the new function?

A. (f(x)+5)
B. (f(x)-5)
C. (f(x-5))
D. (f(x+5))

4

A function is modeled by:

[ P(t)=120(1.08)^t ]

What does 1.08 represent?

5

If (f(x)=x+3) and (g(x)=2x), what is (g(f(4)))?

Answers

1. 21
2. 5 and -5
3. C
4. 8% growth per time period
5. 14

Common traps

• treating (f(x)) as multiplication;
• reversing horizontal shifts;
• confusing input with output;
• interpreting intercept without units;
• assuming every function is linear.

See the rest of our SAT Math guides.

Official references

• College Board Advanced Math: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced

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