Perfect1600

Digital SAT · Solve it in Desmos

How to Solve Function Notation on Desmos (Digital SAT)

Desmos is ideal here, especially on grid-ins where a hand substitution is easy to slip. Define the function once and evaluate as many inputs as you need; for a reverse question, graph it against the target value.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for function notation

The Desmos steps

  1. 1

    Define the function

    Type the definition, like f(x) = x^2 - 2x, on its own line.

  2. 2

    Ask for the value

    Type f(5) and Desmos returns the output, with no substitution mistake.

  3. 3

    Solve the reverse case

    For f of c equal to a number, graph y = f(x) and y = the number and click the intersection for the input.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Easy example
The function ff is defined by f(x)=x2−2xf(x) = x^{2} - 2x. What is the value of f(5)f(5)?
A
1010
1515
C
2323
D
3535

A: Incorrect. This computes only 2x=2(5)2x = 2(5), ignoring the x2x^{2} term.

B: Correct. Substituting gives 52−2(5)=25−10=155^{2} - 2(5) = 25 - 10 = 15.

C: Incorrect. This subtracts 2 instead of 2x2x: 25−225 - 2.

D: Incorrect. This adds instead of subtracting: 25+1025 + 10.

Explanation

Substitute x=5x = 5: 25−10=1525 - 10 = 15.

Medium example
The function hh is defined by h(x)=4x−5h(x) = 4x - 5. If h(c)=23h(c) = 23, what is the value of cc?
A
4.54.5
77
C
1818
D
2828

A: Incorrect. This subtracts 5 instead of adding it before dividing.

B: Correct. 4c−5=234c - 5 = 23 gives 4c=284c = 28, so c=7c = 7.

C: Incorrect. This computes 23−523 - 5 but forgets to divide by 4.

D: Incorrect. This computes 23+523 + 5 but forgets to divide by 4.

Explanation

Set 4c−5=234c - 5 = 23; add 5 to get 4c=284c = 28, then divide by 4 for c=7c = 7.

Hard example
The function ff is defined by f(x)=x2−6x+2f(x) = x^{2} - 6x + 2. What is the value of f(5)f(5)?
A
−7-7
−3-3
C
2121
D
5757

A: Incorrect. This subtracts the constant: 25−30−225 - 30 - 2.

B: Correct. Substituting gives 25−30+2=−325 - 30 + 2 = -3.

C: Incorrect. This uses 6 instead of 6x6x: 25−6+225 - 6 + 2.

D: Incorrect. This drops the sign on −6x-6x: 25+30+225 + 30 + 2.

Explanation

Substitute x=5x = 5: 25−30+2=−325 - 30 + 2 = -3.

Detailed explanation

Computing each term gives 52=255^{2} = 25, −6(5)=−30-6(5) = -30, and +2+2, so 25−30+2=−325 - 30 + 2 = -3. Mishandling the sign of −6x-6x or the constant produces the other values.

Try it with the calculator

Question 1easy

The function ff is defined by f(x)=8xf(x) = 8x. For what value of xx does f(x)=72f(x) = 72?

Question 2easy

The function ff is defined by f(x)=2x−3f(x) = 2x - 3. What is the value of f(5)f(5)?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

Start the free diagnostic

Desmos questions

Can Desmos solve function notation questions on the SAT?

Desmos is ideal here, especially on grid-ins where a hand substitution is easy to slip. Define the function once and evaluate as many inputs as you need; for a reverse question, graph it against the target value.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master function notation, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

See the full methodTake the free diagnostic