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IB Mathematics: Applications and Interpretation HL · 12.5: Functions · 12.5.10

Functions: exam review Exam review: second paper Answers

Section A: short questions

    1. (a) \(2 x^{2} + 1\)
    2. (b) \(4 x^{2} - 20 x + 28\)
    3. (c) \(x\ge3\)
    1. (a) (1.8, 6.05)
    2. (b) (6.05, 1.8)
    3. (c) 1.8
    4. (d) They are inverse functions: the graph of g is the reflection of the graph of f in y = x
    1. (a) \(\frac{2 x}{7} + \frac{4}{7}\)
    2. (b) f(f⁻¹(x)) = x
    1. (a) k = 3.33
    2. (b) \(f(x)\ge −15.5\)
    3. (c) 4
    1. (a) x = 5 and y = 7
    2. (b) b = 2, c = 6
    3. (c) \(f(x)\ne 7\)
    1. (a) (0.9, 7.94)
    2. (b) B is (7.94, 0.9); mirror line y = x
    3. (c) 0.900
    1. (a) t = −5 and S = 720
    2. (b) The number of subscribers gets close to 720 but never reaches or passes it.
    3. (c) 480
    1. (a) C(30) = 16.70; n = 40
    2. (b) 0 ≤ n ≤ 80, n a whole number; 3.2 ≤ C ≤ 39.20
    3. (c) x ≥ −2; g(x) ≥ −1
    4. (d) −3 ≤ h(x) ≤ 6
    1. (a) p(x) = √x and q(x) = x² + 5
    2. (b) x = −3 or x = 3
    3. (c) x = −0.264 or x = 1.26
    1. (a) y = x² − x − 6
    2. (b) y = 3(2^x) + 4; asymptote y = 4
    3. (c) vertical stretch, scale factor 2, and reflection in the x-axis; horizontal stretch, scale factor 1/3, and reflection in the y-axis; image (−2, 8)

Section B: long questions

    1. (a) 12
    2. (b) 0
    3. (c) x = −1, 0, 4, 5
    4. (d) \(x^{4} - 8 x^{3} + 11 x^{2} + 20 x\)
    5. (e) \(f(x)\ge −4\)
    1. (a) (i) (3, −7), (ii) (−2, −5), (iii) (3, −15)
    2. (b) (1, −20)
    3. (c) \(y=2(x+2)^2-5\): vertical stretch, scale factor 2; translation 2 units left and 5 units down
    4. (d) translation 6 right, then horizontal stretch with scale factor 1/3; or horizontal stretch with scale factor 1/3, then translation 2 right
    5. (e) \(y=-x^{2} - 6 x - 4\)
    1. (a) As t increases, k^(−t) tends to 0, so D tends to a; a is the total distance
    2. (b) b = −25
    3. (c) k = 1.47
    4. (d) 5.35 km
    5. (e) 6.55 hours
    1. (a) shown
    2. (b) 6.93
    3. (c) (7.90, 12.9)
    1. (a) As t increases, \(k^{-t}\) tends to 0, so D tends to a; a = 40
    2. (b) b = −40
    3. (c) k = 1.60
    4. (d) 9.77 km
    5. (e) 9 hours
    1. (a) y-intercept −48; zeros x = 2 and x = 12; vertex (7, 50); the greatest profit is 50 euros, made at a price of 7 euros
    2. (b) T = 20; the temperature falls towards but never goes below 20 °C (the temperature of the studio); 89.6 °C
    3. (c) a = 240; N = 48; as p tends to 0, N grows without limit, so the graph never meets the N-axis (a price of 0 gives no value of N)
    4. (d) C(0) = 15; n = −6 and C = 5
    5. (e) Exponential; each value is 3 times the one before
    1. (a) f⁻¹(x) = e^(x − 3); domain: all real x; range: f⁻¹(x) > 0
    2. (b) g⁻¹(x) = −2 + √(x + 4); domain x ≥ −4; range g⁻¹(x) ≥ −2
    3. (c) x = 0.0525 or x = 4.51

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