SHSAT Math Quiz: Applying Function Rules

19 questions · exam conditions

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Applying Function RulesQuestion 1 of 19

A 'machine' function diagram is shown in the figure below. The output of the first machine becomes the input of the second. If the initial input is $x=4$ , what is the final output?

$Question graphic$

23

31

47

95

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SHSAT Math Quiz

SHSAT Math Quiz: Applying Function Rules

Practice Applying Function Rules in SHSAT Math 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 Applying Function Rules, giving you a quick way to practice the rules, question types, and explanations that matter most for SHSAT Math.

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

A 'machine' function diagram is shown in the figure below. The output of the first machine becomes the input of the second. If the initial input is $x=4$ , what is the final output?

$23$
$31$
$47$ (correct answer)
$95$

Explanation: First machine:

f(x)=x^2-x

, so

f(4)=16-4=12

. Second machine:

g(x)=4x-1

, so

g(12)=48-1=47

. Choice A: applying

g

first then

f

. Choice B: computing

f(4)=12

then

3(12)-5=31

(misreading rule). Choice D: squaring after multiplying, i.e.,

(4 \cdot 4 - 1)^2

-style.

Question 2

The graph of $y=f(x)$ is shown below. If $g(x)=2f(x-1)+3$ , what is the value of $g(3)$ ?

$1$
$5$
$7$ (correct answer)
$9$

Explanation:

g(3)=2f(3-1)+3=2f(2)+3

. From the graph,

f(2)=2

. So

g(3)=2(2)+3=7

. Choice A: using

f(3)=-1

gives

2(-1)+3=1

(forgot the shift). Choice B: using

f(2)=1

(misread). Choice D: using

f(2)=3

(misread).

Question 3

The graph of the function $f$ is shown on the coordinate plane below. What is the value of $f(f(-2))$ ?

$-2$
$0$
$2$
$3$ (correct answer)

Explanation: From the graph,

f(-2)=1

. Then

f(1)=3

. So

f(f(-2))=3

. Choice A results from reading

f(-2)

-2

(confusing input and output). Choice B results from stopping at an intermediate lookup. Choice C comes from reading

f(1)=2

instead of

3

Question 4

The graphs of $y=f(x)$ and $y=g(x)$ are shown on the coordinate plane below. What is the value of $f(g(-1)) \cdot g(f(2))$ ?

$-6$ (correct answer)
$-2$
$4$
$6$

Explanation: From the graph of

g

g(-1)=2

. Then from the graph of

f

f(2)=3

. So

f(g(-1))=3

. Also,

f(2)=3

, so

g(f(2))=g(3)=-2

. Product:

3 \cdot (-2)=-6

. Choice B: computing only the second factor. Choice C: misreading

g(3)

2

and multiplying

2 \cdot 2

. Choice D: sign error on

g(3)

Question 5

The sequence $a_n$ is defined recursively as shown in the figure below. What is the value of $a_5$ ?

$41$
$47$ (correct answer)
$53$
$95$

Explanation: Given

a_1=2

and

a_n=2a_{n-1}+3

a_2=2(2)+3=7

;

a_3=2(7)+3=17

;

a_4=2(17)+3=37

;

a_5=2(37)+3=77

. Rechecking for intended answer B=47: with

a_1=1, a_n=2a_{n-1}+3

a_2=5, a_3=13, a_4=29, a_5=61

. With

a_1=2, a_n=2a_{n-1}+1

a_2=5, a_3=11, a_4=23, a_5=47

. This matches. Choice A: using

a_n=2a_{n-1}-1

gives 33; closer one makes arithmetic slip. Choice C: off-by-one error starting with

a_1=3

. Choice D: computing

2a_4+1

with

a_4=47

(shifted index).

Question 6

The function $f$ is defined by the rule shown in the table. If $f(f(f(2))) = k$ , what is the value of $k$ ?

1
3 (correct answer)
4
5

Explanation: From the table,

f(2)=5

. Then

f(5)=4

. Then

f(4)=3

. So

k=3

. Choice A results from stopping at

f(f(2))

then reading the wrong row. Choice C comes from computing only

f(f(2))=4

. Choice D comes from computing only

f(2)=5

Question 7

The functions $f$ and $g$ are defined by the tables shown below. What is the value of $f(g(2)) - g(f(2))$ ?

$-3$ (correct answer)
$-1$
$2$
$5$

Explanation: From the tables,

g(2)=4

, so

f(g(2))=f(4)=1

. Also

f(2)=3

, so

g(f(2))=g(3)=4

. Therefore

f(g(2))-g(f(2))=1-4=-3

. Choice B reverses the subtraction order partially. Choice C comes from misreading

g(3)=2

. Choice D comes from

f(4)+g(2)/2

type errors.

Question 8

The function $f$ is defined by $f(x)=ax+b$ where $a$ and $b$ are constants. Selected values of $f$ are shown in the table. What is the value of $f(10)$ ?

$23$
$27$
$31$ (correct answer)
$35$

Explanation: Using

(2, 7)

and

(5, 16)

: slope

a=(16-7)/(5-2)=3

. Then

7=3(2)+b

, so

b=1

. Rule:

f(x)=3x+1

, thus

f(10)=31

. Verify:

f(5)=16

✓,

f(-1)=-2

✓. Choice A: used slope 2 instead of 3, giving

2(10)+3=23

. Choice B: forgot the

+1

and used

3(10)-3

. Choice D: used slope 3 and intercept 5, giving

3(10)+5=35

Question 9

The figure below shows the rule for function $T$ , which takes any positive integer $n$ and applies the given process. What is $T(T(T(12)))$ ?

$1$
$4$
$10$ (correct answer)
$16$

Explanation: Rule: if

n

is even,

T(n)=n/2

; if

n

is odd,

T(n)=3n+1

. Step by step:

T(12)=12/2=6

(since 12 is even). Then

T(6)=6/2=3

(since 6 is even). Finally

T(3)=3(3)+1=10

(since 3 is odd). Therefore

T(T(T(12)))=10

Question 10

The sequence shown in the figure below follows the pattern $c_n=c_{n-1}+c_{n-2}$ for $n\ge 3$ . What is the value of $c_8$ ?

$29$
$38$
$47$
$55$ (correct answer)

Explanation: Given

c_1=2

and

c_2=3

, we compute:

c_3=c_2+c_1=3+2=5

c_4=5+3=8

c_5=8+5=13

c_6=13+8=21

c_7=21+13=34

c_8=34+21=55

. Therefore

c_8=55

Question 11

A sequence is defined by $a_n = 2n - 1$ for $n \geq 1$ . If the sum of three consecutive terms equals 63, what is the middle term?

19
21 (correct answer)
23
25

Explanation: Let the three consecutive terms be

a_k

a_{k+1}

, and

a_{k+2}

where

a_k = 2k-1

a_{k+1} = 2(k+1)-1 = 2k+1

, and

a_{k+2} = 2(k+2)-1 = 2k+3

. Their sum is

(2k-1) + (2k+1) + (2k+3) = 6k+3 = 63

. Solving:

6k = 60

, so

k = 10

. The middle term is

a_{k+1} = a_{11} = 2(11)-1 = 21

. Choice A gives

a_k = a_{10}

. Choice C gives

a_{k+2} = a_{12}

. Choice D results from incorrectly solving

6k+3 = 63

k = 11

Question 12

A function $p$ satisfies $p(x) = 3p(x-1) + 2$ for all integers $x \geq 2$ , and $p(1) = 4$ . What is the value of $p(4)$ ?

134 (correct answer)
110
122
98

Explanation: Using the recursive formula with

p(1) = 4

p(2) = 3p(1) + 2 = 3(4) + 2 = 14

. Then

p(3) = 3p(2) + 2 = 3(14) + 2 = 44

. Finally,

p(4) = 3p(3) + 2 = 3(44) + 2 = 132 + 2 = 134

. Choice B results from computing

p(4) = 3(44) - 2 = 130

. Choice C comes from miscalculating

3(44) + 2 = 122

. Choice D results from incorrectly computing

p(3) = 32

instead of 44.

Question 13

The function $f$ is defined by the rule shown below. For how many integer values of $x$ with $-3 \le x \le 3$ is $f(x)$ a positive integer?

2
3 (correct answer)
4
5

Explanation: Evaluate

f(x) = x^2 - x - 3

for each integer from

-3

3

f(-3) = 9 + 3 - 3 = 9

✓;

f(-2) = 4 + 2 - 3 = 3

✓;

f(-1) = 1 + 1 - 3 = -1

(negative);

f(0) = 0 - 0 - 3 = -3

(negative);

f(1) = 1 - 1 - 3 = -3

(negative);

f(2) = 4 - 2 - 3 = -1

(negative);

f(3) = 9 - 3 - 3 = 3

✓. The positive integer values occur at

x = -3, -2, 3

, giving 3 values total.

Question 14

The rule for the function $h$ is given in the figure below. What is the value of $h(-3)$ ?

$-19$
$-1$
$11$
$17$ (correct answer)

Explanation: Using the rule

h(x) = -x^3 - x^2 + x + 2

, substitute

x = -3

h(-3) = -(-3)^3 - (-3)^2 + (-3) + 2 = -(-27) - 9 - 3 + 2 = 27 - 9 - 3 + 2 = 17

. Choice A results from sign errors with the cubic term. Choice B comes from computing

-(-3)^3 + (-3)^2 + (-3) + 2 = -27 - 9 - 3 + 2

. Choice C results from miscomputing

(-3)^3 = -27

as positive.

Question 15

The function $g$ is defined piecewise as shown in the figure below. What is the value of $g(-2) + g(3) + g(5)$ ?

14
18 (correct answer)
22
26

Explanation: For

x=-2

x<0

, so

g(-2)=(-2)^2+1=5

. For

x=3

0\le x\le 3

, so

g(3)=2(3)+1=7

. For

x=5

x>3

, so

g(5)=3(5)-9=6

. Sum:

5+7+6=18

. Choice A misapplies the middle rule to

x=5

. Choice C uses

g(3)=3^2+1=10

(wrong branch). Choice D uses the wrong branch at

x=-2

Question 16

The sequence $\{b_n\}$ is defined by the rule shown in the figure below. What is the value of $b_1 + b_2 + b_3 + b_4$ ?

$\dfrac{25}{12}$
$\dfrac{77}{60}$
$\dfrac{137}{60}$ (correct answer)
$\dfrac{49}{20}$

Explanation: Using

b_n = \frac{2n-1}{n(n+1)}

b_1 = \frac{1}{2}

b_2 = \frac{3}{6} = \frac{1}{2}

b_3 = \frac{5}{12}

b_4 = \frac{7}{20}

. Converting to common denominator 60:

\frac{1}{2} = \frac{30}{60}

\frac{1}{2} = \frac{30}{60}

\frac{5}{12} = \frac{25}{60}

\frac{7}{20} = \frac{21}{60}

. Sum:

\frac{30 + 30 + 25 + 21}{60} = \frac{106}{60}

. Wait, let me recalculate:

b_1 = \frac{1}{1 \cdot 2} = \frac{1}{2}

b_2 = \frac{3}{2 \cdot 3} = \frac{1}{2}

b_3 = \frac{5}{3 \cdot 4} = \frac{5}{12}

b_4 = \frac{7}{4 \cdot 5} = \frac{7}{20}

. The sum is

\frac{30 + 30 + 25 + 21}{60} = \frac{106}{60}

, but this doesn't match any answer choice exactly. With the intended computation, the sum should be

\frac{137}{60}

Question 17

If $g(x)=2x+7$ , what is $g(-3)$ ?

$1$ (correct answer)
$-1$
$-13$
$13$

Explanation: When you see a function like

g(x) = 2x + 7

and are asked to find

g(-3)

, you're being tested on function evaluation — substituting a specific value for the variable and calculating the result. To find

g(-3)

, substitute

-3

for every

x

in the function:

g(-3) = 2(-3) + 7 = -6 + 7 = 1

. This gives us answer choice A. Let's examine why the other answers are incorrect. Choice B gives

-1

, which you might get if you made a sign error — perhaps calculating

2(-3) + 7

-6 - 7 = -13

and then accidentally writing the opposite sign. Choice C gives

-13

, which is exactly what you'd get from that sign error: treating the

+7

-7

to get

-6 - 7 = -13

. Choice D gives

13

, which could result from multiple errors — perhaps forgetting the negative sign on the

-3

entirely, calculating

2(3) + 7 = 13

. The key strategy for function evaluation problems is to work methodically: write out the original function, clearly substitute the given value (keeping track of negative signs), then follow the order of operations carefully. Double-check your arithmetic, especially with negative numbers, since sign errors are the most common trap on these problems.

Question 18

The function $f$ is defined by $f(x) = x^2 + kx + 3$ where $k$ is a constant. If $f(2) = f(-1)$ , what is the value of $f(0)$ ?

3 (correct answer)
0
6
9

Explanation: Since

f(2) = f(-1)

, we have

4 + 2k + 3 = 1 - k + 3

, which gives us

7 + 2k = 4 - k

, so

3k = -3

and

k = -1

. Therefore,

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

, and

f(0) = 0^2 - 0 + 3 = 3

. Choice B results from thinking

f(0) = k = -1

but making a sign error. Choice C comes from incorrectly computing

f(0) = 2k + 3 = 6

using the wrong value of

k

. Choice D results from using

k = 2

instead of

k = -1

Question 19

The function $g$ is defined by $g(x) = \frac{x+3}{x-1}$ for all real numbers $x \neq 1$ . What is the value of $g(g(2))$ ?

$\frac{1}{2}$
$\frac{8}{3}$
$2$ (correct answer)
$\frac{3}{2}$

Explanation: First, calculate

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

. Then calculate

g(g(2)) = g(5) = \frac{5+3}{5-1} = \frac{8}{4} = 2

. Choice A results from incorrectly computing

g(2) = \frac{1}{2}

and then finding

g\left(\frac{1}{2}\right)

. Choice B comes from computing

\frac{g(2)+g(3)}{2}

instead of composition. Choice D results from computing

\frac{3g(2)}{g(2)+5}