Unit 8
9 marks · NormalThis question is about functions and probability. Answer each part, using reasoning and showing your methods.
A function is defined by .
Explain why is a function.
What is the domain and range of if can be any real number?
A bag contains 2 red, 1 blue, and 1 green ball. A ball is selected at random, noted, replaced, and then a second ball is selected at random.
Draw a probability tree diagram to show the outcomes for both selections.
Calculate the probability that both balls selected are red.
Calculate the probability that at least one of the balls selected is green.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a(i)) f(x) is a function because each input x gives exactly one output 2x+3. · must show each input maps to one output, definition of a function
(a(ii)) Domain: all real numbers; Range: all real numbers · must show domain: real numbers, range: real numbers
(b(i)) — · must show tree diagram with all branches, correct probabilities on branches
(b(ii)) 1/4
(b(iii)) 7/16
Show mark scheme and worked solution
Marking points
B1 (a(i)) · 1 mark · States that each input has exactly one output (definition of function).
B2 (a(i)) · 1 mark · Properly clarifies why f(x) is a function in terms of mapping rule or one-to-one/many-to-one.
B3 (a(ii)) · 1 mark · Correctly gives the domain as all real numbers.
A1 (a(ii)) · 1 mark · Correctly gives the range as all real numbers.
M1 (b(i)) · 2 marks · Draws a tree diagram with correct outcomes and correct probabilities on all branches.
M2 (b(ii)) · 1 mark · Correct multiplication of probabilities for 'both red' path.
A2 (b(ii)) · 0 marks · (Awarded only for correct value, see M2)
M3 (b(iii)) · 1 mark · Identifies and sums appropriate probabilities for (at least one green) OR uses complementary probability.
A3 (b(iii)) · 1 mark · Correct answer (7/16) for probability at least one green ball.
Worked solution
a(i) is a function because for each input x, it gives exactly one output (2x+3). There is only one possible result for any value of x, so the mapping from x to 2x+3 is well-defined and unique.
a(ii) The domain of is all real numbers: any real number can be substituted for x. The range is also all real numbers because as x takes all real values, 2x + 3 covers all values from to .
b(i) The probability tree diagram will have branches for the first ball (Red, Blue, Green) and, for each, branches for the second ball (Red, Blue, Green).
- Probability of picking Red on 1st selection: 2/4 = 1/2. Blue: 1/4. Green: 1/4
- Probabilities on 2nd selection are the same (since replacement): Red: 1/2, Blue: 1/4, Green: 1/4
Diagram must show all 9 possible outcomes with correct probabilities.
b(ii) Probability both red = P(Red on 1st) × P(Red on 2nd) = (1/2) × (1/2) = 1/4
b(iii) Probability at least one green = 1 − P(no greens) P(no greens) = P(both not green) = (prob not green) × (prob not green) = (3/4) × (3/4) = 9/16 So, probability at least one green = 1 − 9/16 = 7/16