Question 32 of 35intermediate🔧 ApplyShort Answer3 marks

Let f={(1,1),(2,3),(0,−1),(−1,−3)}f = \{(1,1), (2,3), (0,-1), (-1,-3)\} be a function from Z\mathbb{Z} to Z\mathbb{Z} defined by f(x)=ax+bf(x) = ax + b, for some integers a,ba, b. Determine a,ba, b.

Correct Answer

a = 2, b = -1

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 8 | (Chapter: 17)
For More Understanding

Explanation

This question is identical to Example 20 in the textbook context. The function is given as f(x) = ax + b, and we need to find integers a and b using the ordered pairs provided. By substituting values from the given pairs into the function, we can solve for a and b.

Solution Steps

  1. Step 1: Since f(x) = ax + b and (0, -1) ∈ f, we have f(0) = a(0) + b = -1. This gives b = -1.

  2. Step 2: Since (1, 1) ∈ f, we have f(1) = a(1) + b = 1. Substituting b = -1, we get a - 1 = 1, which gives a = 2.

  3. Step 3: Verification: f(2) = 2(2) - 1 = 3 ✓ and f(-1) = 2(-1) - 1 = -3 ✓. All ordered pairs satisfy f(x) = 2x - 1.

  4. Step 4: Therefore, a = 2 and b = -1.

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