Desmos for the SAT · Technique 7 of 23

Evaluate a Function, and Find Its Input

Want its value at an x? Type it. Want the x that gives a value? Click where two graphs cross.

SolvesFunction notationFind f(a)Find x when f(x) = kComposition f(g(x))

Watch the lesson

The whole move worked on one real SAT question, narrated.

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Both directions on one real SAT question: f(4) is typing, and going in reverse is a crossing.

Function notation looks harder than it is. $f(4)$ means: put 4 in. Desmos does it directly: define the function on one line, then type $f(4)$ on the next and the value appears. No re-typing the expression with numbers swapped in, which is where the arithmetic slips happen.

These questions come in two kinds, and the difference is which half they hand you. Either they define the function and want its value at a particular x, which is $f(4)$ , and that is one line of typing. Or they define the function, give you a value, and want the x that produces it, which is $f(x) = 48$ , and that one is a crossing: graph the function, graph the value as a horizontal line, and click where they meet.

The second kind, where they give you the value and want the x, is where the obvious move fails. Typing $f(x) = 48$ under a function you have already defined does not solve anything. Desmos reads it as a second definition of f. The new row looks fine, and it even draws a horizontal line at $y = 48$ . The damage shows up on the row that uses f: $f(4)$ stops returning a value and reports "You have defined f in more than one place". The next section is what to type instead.

Nova

f(4) means: put 4 in. Define the function once in Desmos and you can ask it for any value, including f(g(1)), without re-typing anything. That part is easy. The part worth slowing down for is when they give you the value and want the x, because the obvious move there does not work.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps undoing the function step by step.

The question

$f(x) = 8\sqrt{x}$

For what value of x does $f(x) = 48$ ?

The algebra way

Set the expression equal to the target: 48 = 8 sqrt(x).
Divide both sides by 8: 6 = sqrt(x).
Square both sides: x = 36.
Three steps, and the squaring has to happen on both sides.

The Desmos way

Type f(x) = 8 sqrt(x).
Type y = f(x) and y = 48.
Click the crossing and read $x = 36$ .

The payoff: The algebra needs you to undo each operation in the right order. The graph does not care what the operations were: the crossing is the answer whatever f happens to be.

Quick reference

Which direction is the question asking?

The question says	What you type	What you read
What is $f(4)$ ?	`f(x) = ...` then `f(4)`	The value Desmos prints
What is $f(g(1))$ ?	both definitions, then `f(g(1))`	The value Desmos prints
For what x does $f(x) = 48$ ?	`y = f(x)` and `y = 48`	The x-coordinate of the crossing
The answer choices are fractions	the same as above	Click the fraction toggle on that line

Step by step

The whole move, all 6 steps

Type the function once, using its own name: f(x) = .... Desmos draws the curve and remembers what f means, so you can call it by name on any line below.
Want its value at an x? Type f(4) on the next line and read the number. Any input works, including another function: f(g(1)) works the inside out for you.
If the value comes back as a decimal and the choices are fractions, click the fraction toggle at the left of that line.
Want the x that gives a value, and it is multiple choice? Try the choices: type f(6), then f(8), and stop when one returns the value you were given. Usually the fastest route.
Want the x and there are no choices to try? Type y = f(x) on one line and y = (the value) on the next, so both are graphs.
Click inside either equation box to reveal the gray dots, then click the crossing. Its x-coordinate is the input you were asked for.

The question

The function f is defined by $f(x) = 8\sqrt{x}$ .

For what value of x does f(x) = 48?

Step 1 of 5

Type the function once, exactly as the question gives it: f(x) = 8 sqrt(x). Remember the right arrow after the x to get back out of the root.

In the calculator: f(x)=8sqrt(x)

Desmos draws the curve, and it also remembers the name. The name is the useful part: from here you can ask for f of anything on a line of its own.

Now try it on a real SAT question

Head or scratch paper. You know $\sqrt{144}$ . This is one square root and one addition, and you will finish it before the calculator has focus. Knowing which ones not to type is worth as many points as knowing how to type them.

The function f is defined by $f(x) = 4 + \sqrt{x}$ .

What is the value of f(144)?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

Straight from Desmos

Desmos Help Center: Functions and Function NotationDefining a function, calling it, and composing two of them.

The Desmos trap: when to close the calculator

Never type f(x) = 48 under a function you have already defined. Desmos reads it as a second definition of f, not a solve. The new row draws a line at y = 48 and looks fine; the error, "You have defined f in more than one place", lands on the row that uses f, and f(4) stops returning a value. Deleting a definition is not the fix. Use y = f(x) and y = 48 instead.
For a simple input like $f(2)$ on a small polynomial, doing it in your head is faster than typing a definition and calling it.
Desmos answers in decimals. $f(1/3)$ on a function built from thirds comes back as 2.333..., not $7/3$ , so click the fraction toggle before you match it against the choices.
Read which coordinate the question wants. The crossing gives you a pair, and the answer is almost always the x, since the y is the number the question already gave you.
A question that asks for the value of a constant inside the function, rather than an input or an output, is a sliders question, not this one.

Nova

For something like f(2) on x² + 1, do it in your head. Typing a definition and then asking for a value is two lines of typing to save you one multiplication. Use the calculator when the arithmetic is ugly, when it is a composition, or when you are going backward from a value to an input.

All Desmos techniques

Put this move into real practice

"Evaluate a Function, and Find Its Input" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

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