Symbols & Functions: Level 1
1.
For all numbers \(x\) and \(y\), 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.
For all \(u\) and \(v\), let \(u \nabla v = (u+v) – (u-v)\). What is the value of \(8 \nabla 3\)?
Solution (A): Substitute \(u=8\) and \(v=3\).
\((8+3) – (8-3) = 11 – 5 = 6\).
(Alternatively, \( (u+v) – (u-v) = u+v-u+v = 2v \). So, \(2 \times 3 = 6\).)
3.
Let \(x \text{ # } y = \frac{x+y}{y}\). What is the value of \(10 \text{ # } 2\)?
Solution (C): Substitute \(x=10\) and \(y=2\) into the equation.
\( \frac{10+2}{2} = \frac{12}{2} = 6 \).
4.
Let \( \nabla x = 5x – 4 \). What is the value of \( \nabla 3 \)?
Solution (D): Substitute \(x=3\) into the equation.
\( \nabla 3 = 5(3) – 4 = 15 – 4 = 11 \).
5.
Let \( p \text{ & } q = p^2 – q \). What is the value of \( 4 \text{ & } 10 \)?
Solution (A): Substitute \(p=4\) and \(q=10\).
\( 4^2 – 10 = 16 – 10 = 6 \).
6.
Let \(x \text{ ♣ } y = (x-y)(x+y)\). What is the value of \(5 \text{ ♣ } 3\)?
Solution (B): Substitute \(x=5\) and \(y=3\).
\( (5-3)(5+3) = (2)(8) = 16 \).
(Alternatively, this is the formula for difference of squares, \(x^2 – y^2\). \(5^2 – 3^2 = 25 – 9 = 16\).)
7.
If \( x \text{ ¤ } y = (x-y)^2 \) and \(x=5\), what is the value of \(x \text{ ¤ } (x \text{ ¤ } 4)\)?
Solution (D):
1. First, evaluate the parenthesis with \(x=5\): \( (x \text{ ¤ } 4) = (5 \text{ ¤ } 4) \).
2. \( (5 – 4)^2 = 1^2 = 1 \).
3. Now substitute this result back into the main expression: \( x \text{ ¤ } (1) \).
4. \( (5 \text{ ¤ } 1) = (5 – 1)^2 = 4^2 = 16 \).
8.
For a two-digit number \(ab\), let \( *ab* = a \times b \). What is \( *23* + *41* \)?
Solution (B):
1. \( *23* = 2 \times 3 = 6 \).
2. \( *41* = 4 \times 1 = 4 \).
3. Sum = \( 6 + 4 = 10 \).
9.
If \( x \text{ \$ } y = x+y \) and \(x=3\), for what value of \(y\) does \( x \text{ \$ } y = 7 \)?
Solution (A):
1. Substitute the known values into the equation: \( 3 \text{ \$ } y = 7 \).
2. Use the definition of the symbol: \( 3 + y = 7 \).
3. Solve for y: \( y = 7 – 3 = 4 \).
10.
For a number \(x\), let \( \nabla x = x^2 – 1 \). What is \( \nabla 5 – \nabla 3 \)?
Solution (C):
1. Evaluate \( \nabla 5 \): \( 5^2 – 1 = 25 – 1 = 24 \).
2. Evaluate \( \nabla 3 \): \( 3^2 – 1 = 9 – 1 = 8 \).
3. Find the difference: \( 24 – 8 = 16 \).
Score: 0 / 10