Defining New Operations: Practice

Substitute $a = 2$ and $b = 9$.

$2 \star 9 = 2(2) + 9 - 1 = 4 + 9 - 1 = 12$

Square the first input, then subtract the second.

$4 \diamond 5 = 4^2 - 5 = 16 - 5 = 11$

Square $6$ first, then subtract $10$.

$6 \diamond 10 = 6^2 - 10 = 36 - 10 = 26$

The first input is negative, so $2a = 2(-3) = -6$.

$(-3) \star 4 = 2(-3) + 4 - 1 = -6 + 4 - 1 = -3$

Add the inputs, then add the product $ab$.

$2 \oplus 5 = 2 + 5 + (2)(5) = 7 + 10 = 17$

Substitute $a = 5$ and $b = 1$.

$5 \triangle 1 = 3(5) - 2(1) = 15 - 2 = 13$

With $a = 0$, the term $2a$ is $0$.

$0 \star 6 = 2(0) + 6 - 1 = 0 + 6 - 1 = 5$

Squaring the negative input gives a positive, then subtract $5$.

$(-2) \diamond 5 = (-2)^2 - 5 = 4 - 5 = -1$

Both inputs are $1$, so the product $ab$ is also $1$.

$1 \oplus 1 = 1 + 1 + (1)(1) = 3$

With $b = 0$, only $2a - 1$ remains.

$10 \star 0 = 2(10) + 0 - 1 = 20 - 1 = 19$

Even with equal inputs the rule treats the slots differently.

$6 \triangle 6 = 3(6) - 2(6) = 18 - 12 = 6$

Square the first input, then subtract the second.

$7 \diamond 9 = 7^2 - 9 = 49 - 9 = 40$