Question 35 of 35intermediate🔧 ApplyShort Answer3 marks

Let A={9,10,11,12,13}A = \{9, 10, 11, 12, 13\} and let f:A→Nf: A \rightarrow N be defined by f(n)=f(n) = the highest prime factor of nn. Find the range of ff.

Correct Answer

Range of f = {3, 5, 11, 13}

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 12 | (Page 18)
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Explanation

This question tests understanding of functions and their ranges. The range of a function is the set of all output values (images) of the elements in the domain. Here, we need to find the highest prime factor for each element in set A = {9, 10, 11, 12, 13}. For 9 = 3×33 \times 3, the highest prime factor is 3. For 10 = 2×52 \times 5, the highest prime factor is 5. For 11 (a prime number), the highest prime factor is 11 itself. For 12 = 2×22 \times 2 × 3, the highest prime factor is 3. For 13 (a prime number), the highest prime factor is 13 itself. The range is the set of all these distinct output values.

Solution Steps

  1. Step 1: Find f(9). Since 9 = 3×33 \times 3, the highest prime factor is 3.

  2. Step 2: Find f(10). Since 10 = 2×52 \times 5, the highest prime factor is 5.

  3. Step 3: Find f(11). Since 11 is a prime number, the highest prime factor is 11.

  4. Step 4: Find f(12). Since 12 = 2×22 \times 2 × 3, the highest prime factor is 3.

  5. Step 5: Find f(13). Since 13 is a prime number, the highest prime factor is 13.

  6. Step 6: The range of f is the set of all images: {3, 5, 11, 13}.

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