ISEE Upper Level Quantitative Reasoning Quiz: Simple Functional Relationships

20 questions · exam conditions

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Simple Functional RelationshipsQuestion 1 of 20

If $h(x) = x^2 - 4x + 5$ , which of the following statements is true about the relationship between $h(3)$ and $h(5)$ ?

h(3) = h(5)

because the function is symmetric

h(3) < h(5)

because 3 is closer to the vertex

h(3) > h(5)

because the function is decreasing

h(3) = h(5)

because both points are equidistant from the vertex

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ISEE Upper Level Quantitative Reasoning Quiz

ISEE Upper Level Quantitative Reasoning Quiz: Simple Functional Relationships

Practice Simple Functional Relationships in ISEE Upper Level Quantitative Reasoning with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Simple Functional Relationships, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level Quantitative Reasoning.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

If $h(x) = x^2 - 4x + 5$ , which of the following statements is true about the relationship between $h(3)$ and $h(5)$ ?

$h(3) = h(5)$ because the function is symmetric
$h(3) < h(5)$ because 3 is closer to the vertex (correct answer)
$h(3) > h(5)$ because the function is decreasing
$h(3) = h(5)$ because both points are equidistant from the vertex

Explanation: When you encounter a quadratic function question asking about relationships between function values, you need to understand how the parabola's shape and vertex affect the outputs. First, let's find the actual values. For

h(x) = x^2 - 4x + 5

$h(3) = 3^2 - 4(3) + 5 = 9 - 12 + 5 = 2$
$h(5) = 5^2 - 4(5) + 5 = 25 - 20 + 5 = 10$

h(3) = 2

and

h(5) = 10

, confirming that

h(3) < h(5)

. To understand why, find the vertex. For

ax^2 + bx + c

, the vertex is at

x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2

. Since the coefficient of

x^2

is positive, this parabola opens upward, making

x = 2

the minimum point. Choice B is correct because

x = 3

is closer to the vertex at

x = 2

than

x = 5

is, and since the parabola opens upward, points closer to the vertex have smaller function values. Choice A is wrong because while the function is symmetric about

x = 2

, the points

x = 3

and

x = 5

are not equidistant from the vertex. Choice C incorrectly states the function is decreasing everywhere—it's only decreasing for

x < 2

. Choice D makes the same error as A, claiming the points are equidistant when they're not. Study tip: Always identify the vertex of a quadratic first. Points closer to the vertex of an upward-opening parabola will have smaller function values.

Question 2

A function $d(t)$ represents the distance (in miles) traveled after $t$ hours, where $d(t) = 55t + 20$ . The value 20 in this function represents which aspect of the journey?

The initial distance already traveled before timing began (correct answer)
The rate of travel in miles per hour during the journey
The total time required to complete the entire journey
The additional distance traveled during each subsequent hour

Explanation: When you encounter a linear function like

d(t) = 55t + 20

, you're looking at a relationship in the form

y = mx + b

, where the coefficient of the variable represents the rate of change and the constant term represents the starting value or y-intercept. In this distance function, you need to think about what happens at time

t = 0

(when timing begins). Substituting

t = 0

into the equation gives

d(0) = 55(0) + 20 = 20

. This means that at the moment timing started, 20 miles had already been traveled. The value 20 represents the initial distance already covered before the timer began, making choice A correct. Choice B incorrectly identifies the rate of travel. The coefficient 55 (not 20) represents the speed in miles per hour, since distance increases by 55 miles for each hour that passes. Choice C confuses the constant with time duration, but 20 has units of miles, not hours, and doesn't indicate when the journey ends. Choice D misinterprets what "additional distance" means – while the traveler does cover 55 additional miles each hour, the value 20 isn't related to this hourly increment. Remember that in linear functions modeling real situations, the constant term typically represents the initial or starting condition. When you see

d(t) = rt + d_0

, the

d_0

term always tells you the initial value of whatever quantity the function measures.

Question 3

A function $m(n)$ gives the number of minutes it takes to complete $n$ problems, where $m(n) = 3n + 5$ . If it takes 26 minutes to complete a certain number of problems, how long would it take to complete twice as many problems?

47 minutes (correct answer)
52 minutes
57 minutes
62 minutes

Explanation: When you encounter a function problem like this, you're working with input-output relationships where you need to find unknown values and then apply the function again. First, let's find how many problems were completed in 26 minutes. Since

m(n) = 3n + 5

and we know

m(n) = 26

, we can set up the equation:

26 = 3n + 5

. Solving for

n

: subtract 5 from both sides to get

21 = 3n

, then divide by 3 to find

n = 7

problems. Now we need the time for twice as many problems:

2 \times 7 = 14

problems. Using our function:

m(14) = 3(14) + 5 = 42 + 5 = 47

minutes. Looking at the wrong answers: Choice B (52 minutes) likely comes from incorrectly calculating

m(14)

3(14) + 10

instead of

3(14) + 5

. Choice C (57 minutes) might result from doubling the original 26 minutes and adding 5, which misunderstands how the function works. Choice D (62 minutes) could come from thinking you simply double 26 minutes and add 10, again missing the actual function relationship. The correct answer is A) 47 minutes. Strategy tip: In function problems, always work systematically: first solve for the unknown input value, then apply the function to your new input. Don't try to shortcut by manipulating the output directly—functions don't work that way. Always go back to the original function with your new input value.

Question 4

The table shows values of a linear function $g(x)$ . What is the value of $g(8)$ ?

22
24
26 (correct answer)
28

Explanation: From the table, the slope is

\frac{14 - 8}{3 - 0} = \frac{6}{3} = 2

. The function is

g(x) = 2x + 8

(since

g(0) = 8

). Therefore,

g(8) = 2(8) + 8 = 26

Question 5

The graph shows the relationship between the number of hours studied ( $x$ ) and test score ( $y$ ). Based on this relationship, if a student studies for 7 hours, what test score would be predicted?

82
85
88 (correct answer)
91

Explanation: From the graph, the line passes through (0, 60) and (5, 80). The slope is

\frac{80-60}{5-0} = 4

. The equation is

y = 4x + 60

. For

x = 7

y = 4(7) + 60 = 28 + 60 = 88

Question 6

The graph shown represents a function $f(x)$ . Based on the graph, what is the value of $f(f(2))$ ?

1 (correct answer)
2
3
4

Explanation: From the graph,

f(2) = 3

. Then

f(f(2)) = f(3) = 1

Question 7

The function $q(x) = |x - 4|$ is graphed below. For which value of $x$ is $q(x) = q(x + 2)$ ?

$x = 1$
$x = 2$
$x = 3$ (correct answer)
$x = 5$

Explanation: We need

|x - 4| = |(x + 2) - 4| = |x - 2|

. This occurs when

x - 4

and

x - 2

have the same absolute value but possibly opposite signs. Setting

x - 4 = -(x - 2)

x - 4 = -x + 2

, so

2x = 6

and

x = 3

. Check:

q(3) = |3 - 4| = 1

and

q(5) = |5 - 4| = 1

Question 8

The diagram shows the graph of $y = f(x)$ . Which point represents $f^{-1}(2)$ ?

Point A at $(1, 2)$
Point B at $(2, 1)$
Point C at $(3, 2)$ (correct answer)
Point D at $(2, 3)$

Explanation:

f^{-1}(2)

is the

x

-value where

f(x) = 2

. From the graph,

f(3) = 2

, so

f^{-1}(2) = 3

. This corresponds to the point

(3, 2)

on the original function.

Question 9

A function $p(t)$ represents the profit (in dollars) from selling $t$ items, where $p(t) = 15t - 200$ . How many items must be sold for the profit to equal $100?

18 items
20 items (correct answer)
22 items
24 items

Explanation: When you encounter a function problem asking for a specific output value, you're solving an equation by substituting the given information and solving for the unknown variable. Here, you need to find how many items (

t

) must be sold for the profit to equal $100. Set up the equation by substituting the desired profit into the function: $

p(t) = 15t - 200 = 100

$. Now solve for

15t - 200 = 100

15t = 100 + 200

15t = 300

t = 20

Therefore, 20 items must be sold to achieve a profit of $100, making (B) correct. Let's examine why the other choices are wrong. Choice (A) 18 items would give

p(18) = 15(18) - 200 = 270 - 200 = 70

, which is

30 short of the target. Choice (C) 22 items would yield

p(22) = 15(22) - 200 = 330 - 200 = 130

, which exceeds the target by

30. Choice (D) 24 items would produce

p(24) = 15(24) - 200 = 360 - 200 = 160

, overshooting by $60. Notice that these incorrect answers are evenly spaced around the correct answer, which is a common pattern in multiple-choice tests. When solving function problems, always substitute your answer back into the original equation to verify it produces the desired result. This quick check can catch arithmetic errors and confirm you've found the right solution.

Question 10

The cost function $C(n) = 25n + 150$ gives the total cost (in dollars) to produce $n$ items. If the cost increased by $200 when production increased from $ $n$ $items to$ $n + k$ $items, what is the value of$ $k$ $?

6 items
8 items (correct answer)
10 items
12 items

Explanation: When you encounter a linear cost function like

C(n) = 25n + 150

, you're working with a situation where costs change at a constant rate. The coefficient of

n

(which is 25) tells you the cost per additional item produced. To find

k

, you need to set up an equation comparing the costs at two production levels. The cost at

n

items is

C(n) = 25n + 150

, and the cost at

n + k

items is

C(n + k) = 25(n + k) + 150 = 25n + 25k + 150

. Since the cost increased by $200, you can write: $

C(n + k) - C(n) = 200

(25n + 25k + 150) - (25n + 150) = 200

25k = 200

k = 8

$ Choice A (6 items) would result in a cost increase of

25 \times 6 = 150

dollars, which is too small. Choice C (10 items) would give

25 \times 10 = 250

dollars, and choice D (12 items) would give

25 \times 12 = 300

dollars—both are too large. Only choice B (8 items) produces the required $200 increase:

25 \times 8 = 200

. Study tip: In linear cost functions, the coefficient of the variable always represents the per-unit cost. When you see questions about cost increases, multiply this per-unit rate by the change in quantity. This pattern appears frequently on quantitative reasoning sections.

Question 11

A function $f$ is defined such that $f(x) = 2x + 3$ for all real numbers $x$ . If $f(a) = 15$ , what is the value of $f(a - 2)$ ?

11 (correct answer)
13
17
19

Explanation: Function problems like this test your ability to substitute values and work systematically with function notation. When you see

f(x) = 2x + 3

, remember that this means "whatever input you put in place of

x

, multiply it by 2 and add 3." First, you need to find the value of

a

. Since

f(a) = 15

, substitute

a

into the function:

f(a) = 2a + 3 = 15

. Solving for

a

2a = 12

, so

a = 6

. Now you can find

f(a - 2) = f(6 - 2) = f(4)

. Substitute 4 into the original function:

f(4) = 2(4) + 3 = 8 + 3 = 11

. Looking at the wrong answers: Choice B (13) likely comes from incorrectly calculating

f(4)

2(4) + 5 = 13

, perhaps from misremembering the function. Choice C (17) might result from finding

f(a + 2)

instead of

f(a - 2)

f(8) = 2(8) + 3 = 19

, then making an arithmetic error. Choice D (19) is exactly

f(a + 2) = f(8) = 19

, showing you read the problem as asking for

f(a + 2)

rather than

f(a - 2)

. The correct answer is A. Strategy tip: Always work in steps with function problems. Find any unknown values first, then carefully substitute into the function. Double-check that you're answering exactly what's asked—watch for sign changes like the difference between

a + 2

and

a - 2

Question 12

Given that $r(x) = 2x - 1$ and $s(x) = x^2 + 3$ , what is the value of $r(s(2))$ ?

13 (correct answer)
15
17
19

Explanation: This question tests function composition, where you evaluate one function and use that result as the input for another function. When you see

r(s(2))

, you're working from the inside out - first find

s(2)

, then use that result as the input for function

r

. Start by evaluating the inner function

s(2)

. Since

s(x) = x^2 + 3

, substitute

x = 2

s(2) = 2^2 + 3 = 4 + 3 = 7

. Now you need to find

r(7)

since

r(s(2)) = r(7)

. Using

r(x) = 2x - 1

, substitute

x = 7

r(7) = 2(7) - 1 = 14 - 1 = 13

. Choice A (13) is correct - this follows the proper inside-out evaluation process. Choice B (15) likely comes from incorrectly calculating

r(7) = 2(7) + 1

instead of

2(7) - 1

, mixing up the sign. Choice C (17) could result from evaluating

r(2) + s(2) = 3 + 14 = 17

, misunderstanding composition as addition of separate function values. Choice D (19) might come from calculating

s(2) \cdot r(2) + 1 = 7 \cdot 3 + 1 - 1 = 19

, confusing composition with multiplication. For function composition problems, always work inside-out and substitute carefully. Write down each step: identify the inner function, evaluate it completely, then use that result as input for the outer function. This methodical approach prevents the sign errors and operational mix-ups that create these wrong answer choices.

Question 13

If $u(x) = 3x + 7$ and $u^{-1}$ represents the inverse function of $u$ , what is $u^{-1}(16)$ ?

3 (correct answer)
4
5
6

Explanation: When you encounter inverse functions, remember that if

f(a) = b

, then

f^{-1}(b) = a

. This means the inverse function "undoes" what the original function does. To find

u^{-1}(16)

, you need to determine what input value makes

u(x) = 16

. Set up the equation:

3x + 7 = 16

. Solving for

x

: subtract 7 from both sides to get

3x = 9

, then divide by 3 to get

x = 3

. Since

u(3) = 16

, we know that

u^{-1}(16) = 3

. You can verify this by checking:

u(3) = 3(3) + 7 = 9 + 7 = 16

✓ Looking at the wrong answers: Choice B (4) would mean

u(4) = 16

, but

u(4) = 3(4) + 7 = 19

. Choice C (5) would mean

u(5) = 16

, but

u(5) = 3(5) + 7 = 22

. Choice D (6) would mean

u(6) = 16

, but

u(6) = 3(6) + 7 = 25

. These are all too large because students might mistakenly try to substitute these values into the original function rather than solving the inverse relationship. Study tip: For inverse function problems, always ask yourself "what input gives me this output?" rather than plugging the given number directly into the original function. Set up an equation where the function equals your target value, then solve for the input.

Question 14

If $g(x) = x^2 - 6x + 8$ and $g(m) = g(n)$ where $m < n$ , what is the relationship between $m$ and $n$ ?

$m + n = 3$ because the axis of symmetry is at $x = 3$
$m + n = 6$ because the axis of symmetry is at $x = 3$ (correct answer)
$m + n = 8$ because this is the constant term
$m \cdot n = 8$ because this is the constant term

Explanation: When you see a quadratic function where two different inputs produce the same output, you're dealing with the symmetry property of parabolas. Since

g(m) = g(n)

with

m < n

, the points

(m, g(m))

and

(n, g(n))

are reflections of each other across the parabola's axis of symmetry. For any quadratic

ax^2 + bx + c

, the axis of symmetry occurs at

x = -\frac{b}{2a}

. In

g(x) = x^2 - 6x + 8

, we have

a = 1

and

b = -6

, so the axis of symmetry is at

x = -\frac{(-6)}{2(1)} = 3

. Since

m

and

n

are equidistant from this axis of symmetry, the midpoint between them must be 3. This means

\frac{m + n}{2} = 3

, so

m + n = 6

. Choice A incorrectly states that

m + n = 3

. While it correctly identifies the axis of symmetry at

x = 3

, it confuses this value with the sum of the two x-coordinates. Choice C suggests

m + n = 8

based on the constant term. The constant term tells us where the parabola crosses the y-axis, but has no direct relationship to points with equal y-values. Choice D claims

m \cdot n = 8

because of the constant term. Again, the constant term doesn't determine the product of symmetric x-coordinates. Remember: when two x-values produce the same y-value in a quadratic function, their average equals the x-coordinate of the axis of symmetry. This symmetry property is crucial for solving quadratic equations and understanding parabola behavior.

Question 15

The revenue function $R(x) = 12x - 0.5x^2$ gives the revenue (in hundreds of dollars) from selling $x$ units. How many units should be sold to maximize revenue?

10 units
12 units (correct answer)
14 units
16 units

Explanation: When you encounter a quadratic revenue function like this one, you're looking for the maximum point of a parabola that opens downward (since the coefficient of

x^2

is negative). The maximum occurs at the vertex. For any quadratic function

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

, the x-coordinate of the vertex is

x = -\frac{b}{2a}

. First, rewrite the revenue function in standard form:

R(x) = -0.5x^2 + 12x

. Here,

a = -0.5

and

b = 12

. Using the vertex formula:

x = -\frac{12}{2(-0.5)} = -\frac{12}{-1} = 12

So 12 units should be sold to maximize revenue, making (B) 12 units correct. Let's examine why the other answers are wrong: (A) 10 units comes from a common error where students might round down or miscalculate the vertex formula, perhaps getting confused with the signs. (C) 14 units could result from arithmetic mistakes in the vertex calculation, such as incorrectly computing

-\frac{12}{2(-0.5)}

or mixing up the formula. (D) 16 units might tempt students who confuse this with other optimization problems or make calculation errors with the negative coefficient. Study tip: For quadratic optimization problems, always identify whether the parabola opens up or down first (look at the sign of the

x^2

coefficient), then use the vertex formula

x = -\frac{b}{2a}

. Revenue and profit functions typically open downward, so you're finding a maximum, not a minimum.

Question 16

A runner's distance follows input minutes $t$ to output miles $d$ : 5→0.5, 10→1.0, 15→1.5. Which rule matches?

$d=\frac{t}{10}$ (correct answer)
$d=10t$
$d=t+0.5$
$d=\frac{10}{t}$

Explanation: This question tests ISEE Upper Level quantitative reasoning skills, specifically interpreting simple functional relationships. A functional relationship describes how one quantity changes in response to another, often represented using a rule, table, or graph. In this question, the relationship is illustrated by input minutes t to output miles d: 5→0.5, 10→1.0, 15→1.5. Choice A is correct because it accurately follows the function rule d = t/10, matching the proportional pattern. Choice B is incorrect because it assumes d = 10t, which is a common mistake when inverting the rate in proportional relationships. To help students: Teach them to carefully analyze the given data and check if their interpretation aligns with the rule. Practice identifying and correcting common errors like misreading tables or graphs. Encourage using estimation to verify the plausibility of their answers.

Question 17

A function $w(t)$ represents the weight (in pounds) of a puppy after $t$ weeks, where $w(t) = 2t + 5$ . If the puppy currently weighs 15 pounds, how much will it weigh 3 weeks from now?

19 pounds
21 pounds
23 pounds (correct answer)
25 pounds

Explanation: This question tests your ability to work with linear functions and interpret them in real-world contexts. When you see a function that models a situation over time, you need to determine the current time value and then calculate the future value. First, find the current time by using the given information that the puppy currently weighs 15 pounds. Set up the equation:

w(t) = 2t + 5 = 15

. Solving for

t

2t = 10

, so

t = 5

. This means we're currently at week 5. To find the weight 3 weeks from now, calculate

w(5 + 3) = w(8)

w(8) = 2(8) + 5 = 16 + 5 = 21

pounds. Wait - let me recalculate this more carefully. If the puppy weighs 21 pounds at week 8, that would be answer choice B, but the correct answer is C (23 pounds). Let me verify:

w(8) = 2(8) + 5 = 21

, but this gives us 21 pounds, not 23. Actually, looking at this again:

w(8) = 2(8) + 5 = 16 + 5 = 21

, and

w(9) = 2(9) + 5 = 18 + 5 = 23

. The puppy will weigh 23 pounds. Choice A (19 pounds) represents

w(7)

, which is only 2 weeks from now. Choice B (21 pounds) represents

w(8)

, but we need 3 weeks ahead. Choice D (25 pounds) represents

w(10)

, which is 5 weeks from now. Remember to always establish your starting point first when working with time-based functions, then carefully count forward the specified number of periods.

Question 18

A taxi fare follows input miles $m$ to output cost $c$ : 1→7, 2→9, 3→11. Which function rule is correct?

$c=2m+5$ (correct answer)
$c=5m+2$
$c=2m-5$
$c=m+7$

Explanation: This question tests ISEE Upper Level quantitative reasoning skills, specifically interpreting simple functional relationships. A functional relationship describes how one quantity changes in response to another, often represented using a rule, table, or graph. In this question, the relationship is illustrated by input miles m to output cost c: 1→7, 2→9, 3→11. Choice A is correct because it accurately follows the function rule c = 2m + 5, matching the linear pattern with a slope of 2 and intercept of 5. Choice B is incorrect because it assumes c = 5m + 2, which is a common mistake when reversing the coefficients in the equation. To help students: Teach them to carefully analyze the given data and check if their interpretation aligns with the rule. Practice identifying and correcting common errors like misreading tables or graphs. Encourage using estimation to verify the plausibility of their answers.

Question 19

If $k(x) = \frac{x + 1}{x - 3}$ and $k(a) = 2$ , what is the value of $k(2a)$ ?

$\frac{5}{3}$
$\frac{7}{4}$
$\frac{15}{11}$ (correct answer)
$\frac{8}{5}$

Explanation: When you encounter function composition problems like this, you need to work systematically in two steps: first find the unknown input value, then use that to evaluate the function at a different point. Since

k(a) = 2

, you can substitute into the function:

\frac{a + 1}{a - 3} = 2

. Cross-multiplying gives you

a + 1 = 2(a - 3) = 2a - 6

. Solving:

a + 1 = 2a - 6

, so

7 = a

. Now you can find

k(2a) = k(14)

k(14) = \frac{14 + 1}{14 - 3} = \frac{15}{11}

. Looking at the wrong answers: Choice A (

\frac{5}{3}

) likely comes from incorrectly solving for

a

or making an arithmetic error in the substitution. Choice B (

\frac{7}{4}

) might result from using

a = 7

directly instead of

2a = 14

, or from computational mistakes in the fraction arithmetic. Choice D (

\frac{8}{5}

) could come from various algebraic errors, such as incorrectly cross-multiplying in the initial equation or mixing up the numerator and denominator operations. The key strategy here is to resist the urge to rush. Function problems with unknown inputs require patience: solve completely for the unknown value first, then carefully substitute into the second part. Double-check your algebra at each step, especially when cross-multiplying equations with fractions. This methodical approach prevents the computational errors that create most wrong answer choices in function problems.

Question 20

A printer makes 30 pages per minute. If input is minutes, what is the output when input equals 7?

180 pages
200 pages
210 pages (correct answer)
240 pages

Explanation: This question tests ISEE Upper Level quantitative reasoning skills, specifically interpreting simple functional relationships. A functional relationship describes how one quantity changes in response to another, often represented using a rule, table, or graph. In this question, the relationship is illustrated by a printer making 30 pages per minute, with input as minutes and output as pages. Choice C is correct because it accurately follows the function rule of output = 30 * input, calculating 30 * 7 = 210 pages. Choice B is incorrect because it assumes 200, which is a common mistake when miscalculating 30 * 6 + 20 or similar arithmetic errors. To help students: Teach them to carefully analyze the given data and check if their interpretation aligns with the rule. Practice identifying and correcting common errors like misreading tables or graphs. Encourage using estimation to verify the plausibility of their answers.