Question 31 of 33intermediate🔧 ApplyShort Answer2 marks

Let f={(1,1),(2,3),(0,−1),(−1,−3)}f = \{(1,1), (2,3), (0,-1), (-1,-3)\} be a function from ZZ to ZZ 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 | (Page 17)
For More Understanding

Explanation

This question mirrors Example 20 from the textbook. Since f is a linear function of the form f(x) = ax + b, we can use the given ordered pairs to determine the values of a and b. The point (0, -1) directly gives us the constant term b, and then using point (1, 1), we can solve for a. The values can be verified using other points like (2, 3) and (-1, -3).

Solution Steps

  1. Step 1: Using the point (0, -1), we have f(0) = a(0) + b = b = -1

  2. Step 2: Using the point (1, 1), we have f(1) = a(1) + b = a + b = 1

  3. Step 3: Substituting b = -1 into a + b = 1, we get a + (-1) = 1, which gives a = 2

  4. Step 4: Verification: f(2) = 2(2) - 1 = 3 ✓ and f(-1) = 2(-1) - 1 = -3 ✓

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