Function Rules and Output - ISEE Middle Level Quantitative Reasoning

A function is defined by the rule $g(x) = |2x - 15| + x$ . What is the value of $g(-5)$ ?
Answer: 20 The correct answer is 20. Substitute $x=-5$ into the function: $g(-5) = |2(-5) - 15| + (-5)$ . First, evaluate the expression inside the absolute value bars: $2(-5) - 15 = -10 - 15 = -25$ . The expression becomes $|-25| + (-5)$ . The absolute value of -25 is 25. So the expression is $25 + (-5)$ , which simplifies to $25 - 5 = 20$ .
A baker uses a function $F(p) = \frac{3}{4}(\frac{p}{12})$ to calculate the cups of flour needed for $p$ pancakes. If he uses a different function $S(p) = \frac{1}{3}(\frac{p}{12})$ for the cups of sugar, how much more flour than sugar is needed for 48 pancakes?
Answer: $1 \frac{2}{3}$ cups When you encounter function problems involving real-world applications, focus on substituting the given value and then comparing the results. This question tests your ability to evaluate functions and perform arithmetic with fractions. First, calculate the flour needed using $F(p) = \frac{3}{4}(\frac{p}{12})$ . Substituting $p = 48$ : $F(48) = \frac{3}{4} \times \frac{48}{12} = \frac{3}{4} \times 4 = 3$ cups of flour. Next, calculate the sugar needed using $S(p) = \frac{1}{3}(\frac{p}{12})$ . Substituting $p = 48$ : $S(48) = \frac{1}{3} \times \frac{48}{12} = \frac{1}{3} \times 4 = \frac{4}{3}$ cups of sugar. The difference is: $3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{5}{3} = 1\frac{2}{3}$ cups more flour than sugar. Answer choice (A) $1\frac{2}{3}$ cups is correct. Choice (B) $\frac{5}{4}$ cups likely results from calculation errors in fraction arithmetic. Choice (C) $\frac{5}{12}$ cup suggests confusion with the denominator 12 from the original functions, possibly from incorrectly combining fractions. Choice (D) $4\frac{1}{3}$ cups appears to add the flour and sugar amounts instead of finding their difference. Remember: when working with function problems, substitute carefully, simplify step-by-step, and double-check your fraction arithmetic. Converting mixed numbers to improper fractions can help avoid calculation errors.
A function $h(a, b)$ is defined by the rule $h(a, b) = 3a - b^2$ . What is the value of $h(4, -3)$ ?
Answer: 3 The correct answer is 3. Substitute $a=4$ and $b=-3$ into the function rule: $h(4, -3) = 3(4) - (-3)^2$ . Following the order of operations, calculate the exponent first: $(-3)^2 = (-3) \times (-3) = 9$ . Then, do the multiplication: $3(4) = 12$ . Finally, perform the subtraction: $12 - 9 = 3$ .
A number-processing machine follows a two-step rule. First, it multiplies the input number by -3. Second, it subtracts 7 from the result. If the input number is -5, what is the final output?
Answer: 8 The correct answer is 8. The function rule has two steps. Step 1: Multiply the input by -3. So, $(-5) \times (-3) = 15$ . Step 2: Subtract 7 from the result. So, $15 - 7 = 8$ . The final output is 8.
The function $C(k) = 1.61k$ converts a distance in kilometers, $k$ , to miles. The function $F(m) = 5280m$ converts a distance in miles, $m$ , to feet. Approximately how many feet are in a 10-kilometer race?
Answer: 85,000 The correct answer is 85,000. This requires two steps. First, convert 10 kilometers to miles using $C(k)$ : $C(10) = 1.61 \times 10 = 16.1$ miles. Second, convert 16.1 miles to feet using $F(m)$ : $F(16.1) = 5280 \times 16.1$ . We can approximate this as $5300 \times 16 = 84800$ . The closest answer is 85,000.
What is the sum of the machine's output for an input of 6 and the output for an input of 5?
Answer: 41 The correct answer is 41. First, find the output for each input. The input 6 is even, so use the even rule: $3(6) - 4 = 18 - 4 = 14$ . The input 5 is odd, so use the odd rule: $5^2 + 2 = 25 + 2 = 27$ . Finally, find the sum of the two outputs: $14 + 27 = 41$ .
Commission rule: $c = 5s - 2$ . Find $c$ when $s = 7$ .
Answer: 33 This question tests middle school quantitative reasoning skills, specifically using a function rule to determine output. Function rules allow us to calculate output values by substituting input values into an algebraic expression. In this question, the commission rule c = 5s - 2 is used to find commission c when sales s = 7. Choice A is correct because substituting 7 for s yields c = 5(7) - 2 = 35 - 2 = 33, demonstrating proper function application. Choice C (37) might result from adding 2 instead of subtracting, while Choice D (23) could come from calculating 5(5) - 2. Choice B (17) appears to result from a calculation error. To help students, stress careful attention to operations (subtraction vs. addition) and systematic substitution. Commission problems provide real-world context that makes abstract algebra more meaningful.
The $n$ -th term of a sequence is determined by the function $T(n) = n^2 - 2n + 5$ . What is the 8th term in this sequence?
Answer: 53 The correct answer is 53. To find the 8th term, substitute $n=8$ into the function: $T(8) = 8^2 - 2(8) + 5$ . Following the order of operations, first calculate the exponent: $8^2 = 64$ . Then, perform the multiplication: $2(8) = 16$ . The expression becomes $64 - 16 + 5$ . Now, perform subtraction and addition from left to right: $64 - 16 = 48$ , and $48 + 5 = 53$ .
A function $k(x)$ gives the next term in a sequence and is defined by $k(x) = 3x - 5$ . If the first term of the sequence is 4, what is the fourth term?
Answer: 43 The correct answer is 43. This requires applying the function repeatedly. The first term is 4. The second term is $k(4) = 3(4) - 5 = 12 - 5 = 7$ . The third term is $k(7) = 3(7) - 5 = 21 - 5 = 16$ . The fourth term is $k(16) = 3(16) - 5 = 48 - 5 = 43$ .
Store cost rule: $y = 6x + 1$ . Find $y$ when $x = 3$ .
Answer: 19 This question tests middle school quantitative reasoning skills, specifically using a function rule to determine output. Function rules allow us to calculate output values by substituting input values into an algebraic expression. In this question, the store cost rule y = 6x + 1 is used to find total cost y when x = 3. Choice A is correct because substituting 3 for x yields y = 6(3) + 1 = 18 + 1 = 19, demonstrating proper function application and order of operations. Choice B (18) represents the common error of forgetting to add the constant term, calculating only 6(3). Choice C (10) might result from calculating 3(3) + 1, while Choice D (7) could come from adding 6 + 1. To help students, stress the importance of following each step: multiply first, then add. Store pricing contexts help students connect abstract algebra to everyday situations.
The number of diagonals, $d$ , in a polygon with $n$ sides can be found using the function $d(n) = \frac{n(n-3)}{2}$ . How many diagonals does a heptagon (a 7-sided polygon) have?
Answer: 14 The correct answer is 14. A heptagon has 7 sides, so $n=7$ . Substitute this value into the function: $d(7) = \frac{7(7-3)}{2}$ . First, calculate the value in the parentheses: $7-3=4$ . Then multiply: $7 \times 4 = 28$ . Finally, divide by 2: $28 \div 2 = 14$ .
How many more blocks are in Figure 5 than in Figure 4 of the pattern?
Answer: 17 The correct answer is 17. First, calculate the number of blocks in Figure 5 by substituting $n=5$ into the function: $B(5) = 2(5^2) - 5 = 2(25) - 5 = 50 - 5 = 45$ . Next, calculate the number of blocks in Figure 4 by substituting $n=4$ : $B(4) = 2(4^2) - 4 = 2(16) - 4 = 32 - 4 = 28$ . Finally, find the difference between the two figures: $45 - 28 = 17$ .
If the point $(5, -1)$ is transformed by this rule, what are the coordinates of the new point?
Answer: (-4, 10) The correct answer is (-4, 10). The original point is $(x, y) = (5, -1)$ . The rule for the new point's coordinates is $(y - 3, 2x)$ . To find the new x-coordinate, use the original y-coordinate: $y - 3 = -1 - 3 = -4$ . To find the new y-coordinate, use the original x-coordinate: $2x = 2(5) = 10$ . Therefore, the new point is at (-4, 10).
If the starting number $x_0$ is 20, what is the value of the second output, $x_2$ ?
Answer: 11 The correct answer is 11. This is a two-step process. First, find the first output, $x_1$ , by applying the rule to $x_0 = 20$ : $x_1 = f(20) = \frac{1}{2}(20) + 4 = 10 + 4 = 14$ . Now, use this output as the new input to find $x_2$ : $x_2 = f(14) = \frac{1}{2}(14) + 4 = 7 + 4 = 11$ .
The value $V$ of a collectible item after $t$ years is modeled by the function $V(t) = 50(1.1)^t$ . What is the value of the item after 3 years, rounded to the nearest dollar?
Answer: $67 The correct answer is $67. Substitute $t=3$ into the function: $V(3) = 50(1.1)^3$ . First, calculate $(1.1)^3$ : $1.1 \times 1.1 = 1.21$ , and $1.21 \times 1.1 = 1.331$ . Now, multiply this by 50: $50 \times 1.331 = 66.55$ . Rounded to the nearest dollar, the value is $67.
What is the total bill for a month where 860 MB of data were used?
Answer: $43.00 The correct answer is $43.00. Since 860 MB is greater than 500 MB, the function $B(d) = 25 + 0.05(d - 500)$ applies. First, find the amount of data over 500 MB: $860 - 500 = 360$ MB. Next, calculate the cost for this extra data: $0.05 \times 360 = 18$ . Finally, add this to the base cost of $25: $25 + 18 = 43$ . The total bill is $43.00.
Discount rule: $y = 10x - 5$ . Find $y$ when $x = 4$ .
Answer: 35 This question tests middle school quantitative reasoning skills, specifically using a function rule to determine output. Function rules allow us to calculate output values by substituting input values into an algebraic expression. In this question, the discount rule y = 10x - 5 is used to find the discounted price y when x = 4. Choice A is correct because substituting 4 for x yields y = 10(4) - 5 = 40 - 5 = 35, demonstrating proper function application. Choice B (45) might result from adding 5 instead of subtracting, while Choice D (25) could come from calculating 10(4) - 15. Choice C (5) appears to be an error in calculation or misunderstanding of the function. To help students, emphasize the difference between addition and subtraction in function rules. Real-world contexts like discounts help students understand practical applications of linear functions.
What is the "zip" of the number 38?
Answer: 13 The correct answer is 13. First, divide 38 by 5. $38 \div 5 = 7$ with a remainder of 3. So, the whole number quotient is 7 and the remainder is 3. The rule is to add the quotient (7) and twice the remainder ( $2 \times 3 = 6$ ). The result is $7 + 6 = 13$ .
To convert a temperature from degrees Celsius ( $C$ ) to degrees Fahrenheit ( $F$ ), the function $F(C) = \frac{9}{5}C + 32$ is used. An oven must be preheated to 200°C. What is this temperature in degrees Fahrenheit?
Answer: 392°F The correct answer is 392°F. Substitute $C = 200$ into the function: $F(200) = \frac{9}{5}(200) + 32$ . First, multiply $\frac{9}{5}$ by 200. This is $9 \times (200 \div 5) = 9 \times 40 = 360$ . Then, add 32: $360 + 32 = 392$ . The temperature is 392°F.
Commission rule: $c = 3s + 10$ . Find $c$ when $s = 6$ .
Answer: 28 This question tests middle school quantitative reasoning skills, specifically using a function rule to determine output. Function rules allow us to calculate output values by substituting input values into an algebraic expression. In this question, the commission rule c = 3s + 10 is used to find commission c when sales s = 6. Choice A is correct because substituting 6 for s yields c = 3(6) + 10 = 18 + 10 = 28, demonstrating proper function application. Choice C (18) represents the common error of forgetting to add the constant term, calculating only 3(6). Choice D (16) might result from adding before multiplying, while Choice B (13) could come from various calculation errors. To help students, stress the importance of following each step carefully and checking work. Real-world contexts like commission calculations make abstract concepts more concrete and meaningful.

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