Questions: f: R^2 -> R^2, f(x, y) = (ax-2, by+3b) fonksiyonu sabit fonksiyon olduğuna göre a+b toplamı kaçtır? A) 5 B) 2 C) 1 D) -1 E) -3

Transcript text: \[ \begin{array}{l} f: \mathbb{R}^{2} \mapsto \mathbb{R}^{2}, \\ f(x, y)=(a x-2, b y+3 b) \end{array} \] fonksiyonu sabit fonksiyon olduğuna göre $\boldsymbol{a}+\boldsymbol{b}$ toplamı kaçur? A) 5 B) 2 C) 1 D) -1 E) -3

Solution

Solution Steps

To determine when the function $ f(x, y) = (ax - 2, by + 3b) $ is a constant function, we need both components of the function to be constant, regardless of the input values $ x $ and $ y $. This means that the coefficients of $ x $ and $ y $ must be zero. Therefore, we set $ a = 0 $ and $ b = 0 $. The sum $ a + b $ is then calculated.

Step 1: Determine Conditions for Constant Function

To find when the function $ f(x, y) = (ax - 2, by + 3b) $ is constant, we need both components to be independent of $ x $ and $ y $. This requires that the coefficients of $ x $ and $ y $ are zero.

Step 2: Set Coefficients to Zero

We set the coefficients: \[ a = 0 \quad \text{and} \quad b = 0 \]

Step 3: Calculate the Sum

Now, we calculate the sum $ a + b $: \[ a + b = 0 + 0 = 0 \]