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The modulus function
Questions
1 The modulus function and its graph 2 2 Equations involving modulus 3 3 Inequalities involving modulus 4 4 Challenges 5
Answers
5 The modulus function and its graph 6 6 Equations involving modulus 7 7 Inequalities involving modulus 8 8 Challenges 9
The modulus function and its graph
Q 1 2
Equations involving modulus
Q 1 2
Inequalities involving modulus
Q 1 2 3 4 5 6*
Challenges
Q 1**
1. The modulus function: Graphs
- Sketch the following graphs (a) y = |x + 3| (b) y = | 3 x โ 1 | (c) y = |x โ 5 | (d) y = | 3 โ 2 x| (e) y = 2|x + 1| (f) y = 3|x โ 2 | (g) y = โ 2 | 2 x โ 1 | (h) y = 3| 2 โ 3 x| (i) y = | โ |x||
- Draw sketches of each of the following sets of graphs (a) y = (x โ 1)(x โ 3) and y = |(x โ 1)(x โ 3)| (b) y = 4 โ 3 x โ x^2 and y = | 4 โ 3 x โ x^2 | (c) y = x^2 โ 2 and y = |x^2 โ 2 | (d) y = sin x and y = | sin x| (e) y = (x โ 1)(x โ 2)(x โ 3) and y = |(x โ 1)(x โ 2)(x โ 3)| (f) y = cos 2x and y = | cos 2x| and y = cos | 2 x| (g) y = |x โ 2 | and y = ||x| โ 2 |
3. The modulus function: Inequalities
- Express each of the following inequalities in the form |x โ a| < b: (a) 1 < x < 9 |x โ 5 | < 4 (b) โ 4 < x < 6 |x โ 1 | < 5 (c) โ 3 < x < 8 |x โ 2. 5 | < 5. 5 (d) 2 < x < 11 |x โ 6. 5 | < 4. 5
- Solve the following inequalities: (a) |x + 2| < 4 โ 6 < x < 2 (b) | 3 x + 1| โฅ 2 x โค โ1 or x โฅ (^13) (c) |x โ 2 | โค 2 x + 1 x โฅ (^13) (d) | 2 x โ 3 | > |x โ 4 | x โค โ1 or x โฅ (^73)
- Solve the inequality |x| < 4 |x โ 3 | x < 2 .4 or x > 4
- Solve the equations: (a) x + | 2 x โ 1 | = 3 โ 2 , 43 , (b) 3 + | 2 x โ 1 | = x no solution
- Rewrite the function k(x) defined by k(x) = |x + 3| + | 4 โ x| for the following three cases, without using the modulus in your answer. (a) x < โ3 1 โ 2 x (b) โ 3 โค x โค 4 7 (c) x > 4 2x โ 1 Hence sketch the graph of y = k(x)
6*. Sketch the graph of y = | 2 x โ 3 | + | 5 โ x|. (a) Calculate the y-co-ordinate of the point where the graph cuts the y-axis. 8 (b) Determine the gradient of the graph where x < โ 5 โ 3
4. Inequalities involving modulus: Challenges
1**. Find all the solutions of the equation |x+1|โ|x|+3|xโ 1 |โ 2 |xโ 2 | = x+2 x = โ2 or x โฅ 2
6. The modulus function: Equations
- Draw sketches of each of the following sets of graphs (a) y = (^) xโ^12 and y = | (^) xโ^12 | (b) y = (^1) x โ 1 and y = | (^1) x โ 1 |
- By sketching a graph, or otherwise, find the solution(s) to each of the following equations: (a) | 3 x โ 2 | = 1 x = 13 or 1 (b) | 2 x + 3| = 1 โ x x = โ4 or โ (^23) (c) |x โ 3 | = 2x + 1 x = (^23) (d) |x + 1| = | 2 x โ 3 | x = 23 or 4
7. The modulus function: Inequalities
- Express each of the following inequalities in the form |x โ a| < b: (a) 1 < x < 9 |x โ 5 | < 4 (b) โ 4 < x < 6 |x โ 1 | < 5 (c) โ 3 < x < 8 |x โ 2. 5 | < 5. 5 (d) 2 < x < 11 |x โ 6. 5 | < 4. 5
- Solve the following inequalities: (a) |x + 2| < 4 โ 6 < x < 2 (b) | 3 x + 1| โฅ 2 x โค โ1 or x โฅ (^13) (c) |x โ 2 | โค 2 x + 1 x โฅ (^13) (d) | 2 x โ 3 | > |x โ 4 | x โค โ1 or x โฅ (^73)
- Solve the inequality |x| < 4 |x โ 3 | x < 2 .4 or x > 4
- Solve the equations: (a) x + | 2 x โ 1 | = 3 โ 2 , 43 , (b) 3 + | 2 x โ 1 | = x no solution
- Rewrite the function k(x) defined by k(x) = |x + 3| + | 4 โ x| for the following three cases, without using the modulus in your answer. (a) x < โ3 1 โ 2 x (b) โ 3 โค x โค 4 7 (c) x > 4 2x โ 1 Hence sketch the graph of y = k(x)
6*. Sketch the graph of y = | 2 x โ 3 | + | 5 โ x|. (a) Calculate the y-co-ordinate of the point where the graph cuts the y-axis. 8 (b) Determine the gradient of the graph where x < โ 5 โ 3
