We have learnt that to evaluate the value of an algebraic expression, we substitute a value of the variable in the given expression. Consider the example where the wire is bent to form a rectangle. Here, the area of the rectangle, 10x − x2, is a function of x. Can you interpret this as an input-output process? What value does the expression take when x = 6 cm?
Answer: Yes: the input is the length x, the machine works out 10x − x2, and the output is the area. For x = 6 cm the area is 10(6) − 62 = 24 cm2.
Step-by-step solution
Given: A 20 cm wire is bent into a rectangle of length x cm and width (10 − x) cm; Area = 10x − x2
Idea: A function is a rule that turns each input into exactly one output. Here the rule is “multiply the input by 10, then subtract its square”.
- Input-output view: feed in a length x (input). The machine applies the rule 10x − x2. What comes out (output) is the area of the rectangle. Each length gives exactly one area, so the area is a function of x.1 mark
- Input x = 6: output = 10 × 6 − 62 = 60 − 36 = 24. So the area is 24 cm2.1 mark
- This makes sense: length 6 cm means width 10 − 6 = 4 cm, and 6 × 4 = 24 cm2. Note: 10x − x2 has degree 2, so it is a quadratic function, not a linear one.
The area machine takes the length x as input and outputs 10x − x2; for x = 6 cm the output is 24 cm2.
Check: Width = 10 − 6 = 4 cm; 6 × 4 = 24 cm2 ✓.
Answer to write in the exam
Input: length x; rule: 10x − x2; output: area
At x = 6: 10(6) − 62 = 60 − 36 = 24
∴ Area = 24 cm2
Common mistakes that cost marks
- Working out 10 × 6 − 6 × 2 = 48. The term is x2 = 6 × 6 = 36, not 6 × 2.
- Stopping at 10 × 6 = 60 cm2 and forgetting to subtract x2 = 36.
- Giving the answer in cm instead of cm2.
How this can come in the exam
MCQ (1 mark)
For the function A(x) = 12x − x2, the output when the input is 5 is
- 35
- 50
- 60
- −13
Show answer
(A) 35
12 × 5 − 25 = 60 − 25 = 35.
Short answer (2 marks)
A machine follows the rule “output = 3 × input + 4”. Find the outputs for inputs −2 and 7, and say whether the rule is linear.
Show answer
Input −2: 3(−2) + 4 = −2; input 7: 21 + 4 = 25 (1 mark). The rule 3x + 4 has degree 1, so it is linear (1 mark).Try one yourself
Using the area function 10x − x2 for the 20 cm wire, find the area when x = 7.5 cm, and check it with length × width.
Show answer
75 − 56.25 = 18.75 cm2. Width = 2.5 cm and 7.5 × 2.5 = 18.75 ✓.
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