Symbols & Functions: Level 2
1.
Let \(x \text{ \$ } y = 2x + 3y\). What is the value of \(5 \text{ \$ } 2\)?
Solution (D): Substitute \(x=5\) and \(y=2\) into the equation.
\(x \text{ \$ } y = 2x + 3y\)
\(5 \text{ \$ } 2 = 2(5) + 3(2) = 10 + 6 = 16\).
2.
Let \(x \text{ \$ } y = 3x – y\). If \(k \text{ \$ } 5 = 10\), what is the value of \(k\)?
Solution (B):
1. Substitute the knowns into the equation: \( (k \text{ \$ } 5) = 3k – 5 \).
2. We are given that this equals 10.
3. \(3k – 5 = 10\).
4. \(3k = 15\).
5. \(k = 5\).
3.
Let the function \(x\#\) be defined as \(x\# = ax + 5\). If \(4\# = 17\), what is the value of \(6\#\)?
Solution (D):
1. First, find the constant \(a\) using the given information: \(4\# = 17\).
2. \(a(4) + 5 = 17\).
3. \(4a = 12 \implies a = 3\).
4. The function is \(x\# = 3x + 5\).
5. Now, find \(6\#\): \(3(6) + 5 = 18 + 5 = 23\).
4.
For a three-digit number `abc`, let \(f(abc) = 2^a \times 3^b \times 5^c\). What is the value of \(f(211)\)?
Solution (C):
1. The digits are \(a=2, b=1, c=1\).
2. Substitute these into the function: \(f(211) = 2^2 \times 3^1 \times 5^1\).
3. \(f(211) = 4 \times 3 \times 5 = 12 \times 5 = 60\).
5.
Let \(x \nabla y = x^2 – y\). What is the value of \((3 \nabla 2) \nabla 5\)?
Solution (A):
1. First, evaluate the parenthesis: \((3 \nabla 2)\).
2. \(3 \nabla 2 = 3^2 – 2 = 9 – 2 = 7\).
3. Now substitute this result back: \(7 \nabla 5\).
4. \(7 \nabla 5 = 7^2 – 5 = 49 – 5 = 44\).
6.
Let \(x \text{ Ω } y = (x-y)^2\). If \(k \text{ Ω } 5 = 36\), which of the following could be \(k\)?
Solution (B):
1. Substitute the knowns into the equation: \((k – 5)^2 = 36\).
2. Take the square root of both sides: \(k – 5 = 6\) OR \(k – 5 = -6\).
3. Case 1: \(k – 5 = 6 \implies k = 11\).
4. Case 2: \(k – 5 = -6 \implies k = -1\).
5. Of the options provided, 11 is a possible value for \(k\).
7.
Let \( \nabla x = 3x – 3 \). What is the value of \( \frac{\nabla 5}{\nabla 2} \)?
Solution (D): This is a ratio of two separate function calls.
1. Evaluate the numerator, \(\nabla 5\):
\(\nabla 5 = 3(5) – 3 = 15 – 3 = 12\).
2. Evaluate the denominator, \(\nabla 2\):
\(\nabla 2 = 3(2) – 3 = 6 – 3 = 3\).
3. Calculate the ratio: \( \frac{12}{3} = 4 \).
8.
Let \(x \text{ ♦ } y = x^2 + y^2 – 2xy\). What is the value of \(12 \text{ ♦ } 5\)?
Solution (E):
Method 1 (Substitution):
\(12^2 + 5^2 – 2(12)(5) = 144 + 25 – 120 = 169 – 120 = 49\).
Method 2 (Factoring):
The expression \(x^2 + y^2 – 2xy\) is the same as \((x-y)^2\).
So, \(12 \text{ ♦ } 5 = (12 – 5)^2 = 7^2 = 49\).
9.
Let \(x\# = ax^2 + b\). If \(1\# = 5\) and \(2\# = 11\), what is the value of \(3\#\)?
Solution (C): This is a system of equations.
1. \(1\# = 5 \implies a(1)^2 + b = 5 \implies a + b = 5\).
2. \(2\# = 11 \implies a(2)^2 + b = 11 \implies 4a + b = 11\).
3. Subtract the first equation from the second:
\((4a + b) – (a + b) = 11 – 5 \implies 3a = 6 \implies a = 2\).
4. Substitute \(a=2\) back: \(2 + b = 5 \implies b = 3\).
5. The function is \(x\# = 2x^2 + 3\).
6. Find \(3\#\): \(2(3)^2 + 3 = 2(9) + 3 = 18 + 3 = 21\).
10.
For a positive integer \(x\), let \(x^*\) be the product of the digits of \(x\). What is \((23)^* + (15)^* – (11)^*\)?
Solution (D):
1. Evaluate \((23)^*\): \(2 \times 3 = 6\).
2. Evaluate \((15)^*\): \(1 \times 5 = 5\).
3. Evaluate \((11)^*\): \(1 \times 1 = 1\).
4. Calculate the final expression: \(6 + 5 – 1 = 10\).
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