Hidden functions inside quotients

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Lesson progressPractice problems 0/5

Difficulty: GOAT Mode
Estimated time: 55 minutes
Domains: Advanced Math
Techniques: QuotientsInterpretationCustom-regressionCalculated-columnsFunctionsVerification

What you’ll learn

Identify which function owns the supplied table and which function the question asks about.
Reverse a quotient relationship to recover hidden function values row by row.
Fit the complete quotient model with a custom regression.
Preserve known terms around the hidden function.
Choose the shorter route when one table row already contains the requested input.
Read an intercept, output, slope, or vertex from the correct function.
Verify the recovered function in the original quotient relationship.

Prerequisites

Why this matters on the SAT

Recover the function the question actually asks about

Some hard SAT function questions give a table for one function, define that function using another, and then ask about the function hidden inside the definition. The visible table is only the outer layer.

SAT example

The table shows three values of $x$ and their corresponding values of $g(x)$ , where

g(x)=\frac{f(x)}{x+3}

and $f$ is a linear function.

$x$	$g(x)$
$-27$	$3$
$-9$	$0$
$21$	$5$

What is the $y$ -intercept of the graph of $y=f(x)$ in the $xy$ -plane?

A. $(0,12)$

B. $(0,4)$

C. $(0,-9)$

D. $(0,36)$

Fast Desmos solution

The table contains values of $g$ , but the question asks about the linear function $f$ . Let

f(x)=mx+b.

Enter the table in x_1 and y_1, then fit the complete quotient:

y_1~(mx_1+b)/(x_1+3)

Desmos gives

m=4\qquad\text{and}\qquad b=36.

Therefore,

f(x)=4x+36,

so the $y$ -intercept of $y=f(x)$ is

\boxed{(0,36)}.

The answer is D.

Common mistake:

Reading the quotient’s intercept. Since $g(0)=f(0)/3=12$ , the point $(0,12)$ belongs to $g$ , not $f$ . It is the tempting answer A.

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Compare the quotient curve with the hidden line, then edit f(0) to test another input.

Read the layers before touching the calculator

A hidden function is a function whose values are not supplied directly but appear inside the definition of another function.

g(x)=\frac{f(x)}{x+3},

the layers are:

Question part	Meaning
Table headings	The supplied outputs are values of $g$
Quotient relationship	Each value of $g$ combines $f(x)$ with the known denominator $x+3$
Function family	The statement that $f$ is linear tells you to use $f(x)=mx+b$
Requested target	The $y$ -intercept belongs to $f$ , so you need $f(0)$ or $b$
Restriction	The quotient defining $g$ is undefined at $x=-3$

Try it yourself:

Before entering anything, complete this sentence: The table belongs to ______, but the question asks for ______ of ______.

For the opening problem: The table belongs to $g$ , but the question asks for the $y$ -intercept of $f$ . That one sentence prevents the most common interpretation error.

Check your understanding:

A table lists values of $h(x)$ , where $h(x)=p(x)/(x-4)$ and $p$ is quadratic. The question asks for the minimum value of $p$ . Which function should you fit, and which function owns the minimum?

1. Unwrap the quotient row by row

Multiply by the known denominator

g(x)=\frac{f(x)}{x+c},

then for every allowed input,

f(x)=(x+c)g(x).

This is not a new model. It is the original relationship solved for the hidden value.

For the opening table:

\begin{aligned} f(-27)&=(-27+3)(3)=-72,\\ f(-9)&=(-9+3)(0)=0,\\ f(21)&=(21+3)(5)=120. \end{aligned}

The table for $g$ therefore produces three points on $f$ :

(-27,-72),\quad(-9,0),\quad(21,120).

Those points lie on

f(x)=4x+36.

Let one table formula recover every row

If the supplied inputs are in x_1 and the supplied values of $g$ are in y_1, enter (x_1+3)y_1 as a calculated column. The column displays the corresponding values of $f$ .

This route is especially transparent because you can inspect the recovered points before fitting anything.

Common mistake:

Multiplying only the input by the denominator. The relationship is $f(x)=(x+3)g(x)$ , so multiply the supplied output $g(x)$ by the denominator value.

Check your understanding:

If $q(x)=r(x)/(2x-1)$ and a table row gives $x=3$ and $q(3)=4$ , what is $r(3)$ ?

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Change one supplied g value and watch the recovered f column update, then reset.

2. Fit the full quotient directly

Keep the wrapper around the model

The row-by-row method and the direct regression method use the same mathematics.

The SAT supplies exact function values in these problems. Regression is acting as a numerical solver for the stated relationship, not as permission to accept an approximate best-fit model. Verify that the recovered function reproduces every supplied row.

If $f$ is linear, write

f(x)=mx+b.

Then the quotient relationship becomes

g(x)=\frac{mx+b}{x+c}.

When the table stores $x$ in x_1 and $g(x)$ in y_1, enter y_1~(mx_1+b)/(x_1+c). Use the actual numerical value of $c$ from the problem. The tilde tells Desmos to fit $m$ and $b$ so that the complete quotient matches the table.

Common mistake:

Fitting the table as a line. y_1~mx_1+b treats the table values as outputs of $f$ . They are outputs of $g$ . Even when $f$ is linear, the quotient function $g$ is generally not linear.

Check your understanding:

A table contains values of $h(x)=p(x)/(x-5)$ , where $p(x)=ax^2+bx+c$ . What custom regression preserves the complete relationship?

Choose between the two setups

Use row-by-row reconstruction when:

the denominator is simple;
seeing the hidden points will clarify the problem;
one supplied row may already answer the question; or
hand arithmetic is shorter than fitting.

Use a full quotient regression when:

the hidden function family is stated;
several parameters must be recovered;
the quotient is easy to enter accurately; and
you want to define and evaluate the hidden function immediately after fitting.

Both methods should produce the same hidden function. If they do not, recheck the table labels, signs, denominator, and model family.

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Enter the complete quotient regression, define the hidden line, and verify all three rows.

3. Stop if one row already gives the target

Suppose

q(x)=\frac{r(x)+4}{2x-1},

and a table gives

q(2)=5.

The question asks for $r(2)$ . Substitute only that row:

5=\frac{r(2)+4}{2(2)-1}.

Since the denominator is $3$ ,

\begin{aligned} 15&=r(2)+4\\ r(2)&=11. \end{aligned}

There is no reason to recover the entire linear function $r$ . The requested input already appears in the table.

Try it yourself:

If the same table instead asked for $r(0)$ , would the row at $x=2$ be enough? Explain what additional information you would need.

The row at $x=2$ gives only $r(2)$ . To find $r(0)$ , you would need enough information to determine the linear function, such as another independent row.

This is the quotient version of a broader SAT habit: stop when the requested quantity is known, but do not invent values the data does not determine.

4. Preserve every known term around the hidden function

The hidden function is not always the entire numerator.

Suppose

g(x)=\frac{f(x)+2x-5}{x-3}

and $f$ is linear. Writing $f(x)=mx+b$ gives

g(x)=\frac{mx+b+2x-5}{x-3}.

The direct regression is y_1~(mx_1+b+2x_1-5)/(x_1-3). Every known term stays in its original position. The only fitted parameters are $m$ and $b$ from the hidden linear function.

For row-by-row reconstruction, isolate $f(x)$ :

\begin{aligned} g(x)(x-3)&=f(x)+2x-5\\ f(x)&=g(x)(x-3)-2x+5. \end{aligned}

The matching calculated column is y_1(x_1-3)-2x_1+5.

Common mistake:

Deleting the known terms. The prompt has already supplied $2x-5$ . Keep those terms visible and fit only the coefficients that belong to $f$ .

Check your understanding:

If $h(x)=(p(x)-4x+7)/(x+2)$ and $p(x)=ax+b$ , what direct custom regression should you enter for a table of $h$ values?

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Try the full quotient regression or the calculated-column formula. Both should recover f(x) = 2x + 1.

5. Read the requested feature from the correct function

After the regression, define the hidden function on a new line. If the fit gives $m$ and $b$ , enter f(x)=mx+b.

Then translate the question:

Requested feature of $f$	What to evaluate or read
$y$ -coordinate of the $y$ -intercept	$f(0)$ , which is $b$ for $f(x)=mx+b$
$x$ -coordinate of the $x$ -intercept	Solve $f(x)=0$ or click the hidden graph’s $x$ -intercept
Slope	Read $m$ from the linear model
A value such as $f(7)$	Enter `f(7)`
Vertex of a quadratic	Graph the recovered quadratic or use its fitted coefficients
Minimum or maximum value	Read the vertex’s $y$ -coordinate after checking whether the quadratic opens up or down

The quotient function may have completely different intercepts and extrema. Always name the function in your conclusion: The requested $y$ -intercept of $f$ is $(0,36)$ . That wording forces one final layer check.

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Compare f(0) = 36 with g(0) = 12, then edit the final inputs to explore other features on f.

6. Keep the denominator restriction attached to the quotient

g(x)=\frac{f(x)}{x-2},

then $g$ is undefined at $x=2$ . Do not use a table row at that input, and do not claim that the quotient relationship holds there.

However, the hidden function $f$ may still be defined at $x=2$ . For example, if

f(x)=x^2+x-6=(x-2)(x+3),

then $f(2)=0$ , even though

\frac{f(2)}{2-2}

is undefined. The exclusion belongs to the quotient function $g$ , not automatically to the polynomial $f$ .

This lesson uses restrictions to protect the recovery process. For a complete treatment of holes, asymptotes, and canceled factors, return to Restrictions, piecewise functions, and rational expressions.

Check your understanding:

If $h(x)=p(x)/(x+4)$ and $p$ is a polynomial, which statement is guaranteed: $h(-4)$ is undefined, $p(-4)$ is undefined, or both?

7. Guided example: Recover the hidden line through a larger quotient

Worked example

The table gives two values of $x$ and their corresponding values of

$g(x)$ ,

where

g(x)=\frac{f(x)+2x-5}{x-3}

and

$f$ is a linear function.

$x$	$g(x)$
$1$	$4$
$5$	$-1$

Which choice is the $y$ -coordinate of the $y$ -intercept of the graph of

$y=f(x)$ in the $xy$ -plane?

A
$-\frac72$
B
$-\frac{11}{2}$
C
$-\frac94$
D
$-\frac92$

Step 1

Name the table and target layers

The table supplies values of $g$ , while the question asks for the $y$ -intercept of $f$ . The denominator also excludes $x=3$ from the quotient relationship. Neither supplied row uses that input.

Step 2

Insert the stated function family

Because $f$ is linear, define

f(x)=mx+b.

Substitute that model into the complete quotient:

g(x)=\frac{mx+b+2x-5}{x-3}.

The known terms $2x-5$ remain in the numerator.

Step 3

Fit the complete quotient

With the two table rows ready, type y_1~(mx_1+b+2x_1-5)/(x_1-3). Desmos reports the numerical candidates

m=-0.5\qquad\text{and}\qquad b=-4.5.

Written exactly, these are $-\frac12$ and $-\frac92$ . Verify those exact values in the original relationship before accepting them.

Step 4

Answer from f and verify both rows

The hidden function is

f(x)=-\frac12x-\frac92.

Therefore,

f(0)=-\frac92,

so the $y$ -coordinate of the $y$ -intercept is

\boxed{-\frac92}.

The answer is D.

Verify in the original relationship:

\frac{f(1)+2(1)-5}{1-3}=\frac{-5+2-5}{-2}=4

and

\frac{f(5)+2(5)-5}{5-3}=\frac{-7+10-5}{2}=-1.

Both supplied values of $g$ are reproduced.

Common mistake:

Reporting the quotient’s y-intercept. The question asks for the intercept of $f$ , so evaluate $f(0)$ after recovering $f$ .

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When step 3 asks you to, enter the complete quotient regression, define f, and evaluate f(0).

8. Verify the original relationship

A clean regression output is not the end of the solution. Check that the recovered function recreates the supplied outer values.

For each table row:

evaluate the recovered hidden function;
place that value back into the original numerator;
apply the original denominator;
confirm the supplied outer output; and
confirm the denominator is not zero.

For the opening problem, $f(x)=4x+36$ . At $x=21$ :

\frac{f(21)}{21+3}=\frac{120}{24}=5,

which matches the table.

You do not need to write every check during the SAT. Verify one or two rows if the setup is easy to mistype, and always return to the exact function and feature named in the question.

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Edit one check input, compare the result with the corresponding g value, then reset.

Finish the solution

Recover the hidden y-intercept

Finish the solution

The table gives four values of $x$ and their corresponding values of $g(x)$ , where

g(x)=\frac{f(x)}{x+2}

and $f$ is a linear function.

$x$	$g(x)$
$-6$	$1$
$-4$	$-2$
$1$	$8$
$4$	$6$

What is the $y$ -coordinate of the $y$ -intercept of the graph of $y=f(x)$ ?

First steps

The table is ready. Enter the full quotient regression, define the hidden line, and submit the requested intercept value.

Let $f(x)=mx+b$ .
Preserve the complete relationship: $g(x)=(mx+b)/(x+2)$ .
Fit that relationship to the table.
Define $f$ and evaluate $f(0)$ .

Finish it

Your answer

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Enter the complete quotient model, then define f and evaluate its y-intercept.

Practice problems

SAT practice problems

Name the table function, isolate or model the hidden function, and answer from the layer named in the question. Use a full regression only when it saves work.

Stop at the requested row

Practice problem

The table gives three values of $x$ and their corresponding values of $q(x)$ , where

q(x)=\frac{r(x)+4}{2x-1}

and $r$ is a linear function.

$x$	$q(x)$
$-1$	$-2$
$2$	$5$
$5$	$\frac83$

What is the value of $r(2)$ ?

Your answer

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Inspect the row at the requested input before deciding whether a full regression is necessary.

Recover a hidden x-intercept

Practice problem

The table gives four values of $x$ and their corresponding values of $g(x)$ , where

g(x)=\frac{f(x)}{x+4}

and $f$ is a linear function.

$x$	$g(x)$
$-8$	$-8$
$-6$	$-13$
$0$	$2$
$1$	$1$

What is the $x$ -coordinate of the $x$ -intercept of the graph of $y=f(x)$ ?

Your answer

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Fit the complete quotient or reconstruct the hidden outputs, then find the intercept on f.

Keep the known numerator terms

Practice problem

The table gives three values of $x$ and their corresponding values of $g(x)$ , where

g(x)=\frac{f(x)-3x+2}{x+1}

and $f$ is a linear function.

$x$	$g(x)$
$-5$	$-8$
$1$	$1$
$3$	$-2$

What is the value of $f(-2)$ ?

Your answer

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Preserve -3x + 2 in the numerator while fitting the hidden linear function.

Recover a hidden quadratic

Practice problem

The function $q$ is defined by

q(x)=\frac{p(x)-5}{x+1},

where $p$ is a quadratic polynomial. The table shows three values of $x$ and the corresponding values of $q(x)$ .

$x$	$q(x)$
$2$	$3$
$4$	$7$
$-4$	$-1$

What are the coordinates of the vertex of the graph of $y=p(x)$ in the $xy$ -plane?

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Use a quadratic numerator in the quotient model, then inspect the vertex of p.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Name the layers: Identify the function in the table and the function in the question.
Record the exclusion: A zero denominator invalidates the quotient relationship at that input.
Unwrap: Solve the relationship for the hidden value and calculate it row by row.
Model: Insert the stated hidden function family into the complete quotient before using regression.
Preserve: Keep every known term around the hidden function exactly where the prompt places it.
Stop early: If one row already gives the requested hidden value, do not fit the whole function.
Interpret: Read the requested intercept, output, slope, or vertex from the hidden function, not automatically from the table function.
Verify: Substitute the recovered function into the original quotient and reproduce the supplied outputs.

Return to the Desmos for the SAT course to review earlier techniques or choose another lesson.

Next lesson

Exactness, rounding, and output verification

Decide when a calculator decimal is final, when an exact form is recoverable, and how to match the requested precision.

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Free Practice

What you’ll learn

Prerequisites

Recover the function the question actually asks about

Fast Desmos solution

Read the layers before touching the calculator

1. Unwrap the quotient row by row

Multiply by the known denominator

Let one table formula recover every row

2. Fit the full quotient directly

Keep the wrapper around the model

Choose between the two setups

3. Stop if one row already gives the target

4. Preserve every known term around the hidden function

5. Read the requested feature from the correct function

6. Keep the denominator restriction attached to the quotient

7. Guided example: Recover the hidden line through a larger quotient

Name the table and target layers

Insert the stated function family

Fit the complete quotient

Answer from f and verify both rows

8. Verify the original relationship

Finish the solution

Recover the hidden y-intercept

First steps

Finish it

Practice problems

SAT practice problems

Stop at the requested row

Recover a hidden x-intercept

Keep the known numerator terms

Recover a hidden quadratic

5 practice examples left

Quick recap

Exactness, rounding, and output verification

Free question bank practice problems