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

Functions: exam review Exam review Answers

Section A: short questions

    1. (a) \(3 x^{2} - 1\)
    2. (b) \(9 x^{2} + 12 x + 3\)
    3. (c) \(x\ge\frac{2}{3}\)
    1. (a) (2, 7.39)
    2. (b) (7.39, 2)
    3. (c) 2
    4. (d) They are inverse functions: the graph of g is the reflection of the graph of f in y = x
    1. (a) \(\frac{3 x}{5} - \frac{2}{5}\)
    2. (b) f(f⁻¹(x)) = x
    1. (a) k = 2.33
    2. (b) \(f(x)\ge −8.70\)
    3. (c) 3
    1. (a) x = 2 and y = 5
    2. (b) b = 3, c = 4
    3. (c) \(f(x)\ne 5\)
    1. (a) (0.5, 3.16)
    2. (b) B is (3.16, 0.5); mirror line y = x
    3. (c) 0.500
    1. (a) t = −3 and N = 450
    2. (b) The number of members gets close to 450 but never reaches or passes it.
    3. (c) 300
    1. (a) C(40) = 16.50; n = 60
    2. (b) 0 ≤ n ≤ 120, n a whole number; 2.5 ≤ C ≤ 44.50
    3. (c) x ≥ 4; g(x) ≥ 3
    4. (d) −4 ≤ h(x) ≤ 5
    1. (a) p(x) = 1/x and q(x) = x² + 3
    2. (b) x = −3 or x = 3
    3. (c) x = −5.35 or x = −0.655
    1. (a) y = x² − 4x + 3
    2. (b) y = 4(3^x) − 5; asymptote y = −5
    3. (c) vertical stretch, scale factor 3, and reflection in the x-axis; horizontal stretch, scale factor 1/2, and reflection in the y-axis; image (−2, 18)

Section B: long questions

    1. (a) 3
    2. (b) 4
    3. (c) x = 0, 1, 3, 4
    4. (d) \(x^{4} - 8 x^{3} + 19 x^{2} - 12 x\)
    5. (e) \(f(x)\ge −1\)
    1. (a) (i) (2, 8), (ii) (6, 5), (iii) (2, −10)
    2. (b) (4, 10)
    3. (c) \(y=3(x-2)^2-5\): vertical stretch, scale factor 3; translation 2 units right and 5 units down
    4. (d) translation 6 left, then horizontal stretch with scale factor 1/2; or horizontal stretch with scale factor 1/2, then translation 3 left
    5. (e) \(y=x^{2} + 4 x - 1\)
    1. (a) As t increases, k^(−t) tends to 0, so D tends to a; a is the total distance
    2. (b) b = −18
    3. (c) k = 1.43
    4. (d) 6.17 km
    5. (e) 8.10 hours
    1. (a) shown
    2. (b) 4.90
    3. (c) (5.46, 9.46)
    1. (a) As t increases, \(k^{-t}\) tends to 0, so D tends to a; a = 45
    2. (b) b = −45
    3. (c) k = 1.80
    4. (d) 7.72 km
    5. (e) 8 hours
    1. (a) y-intercept −60; zeros x = 2 and x = 10; vertex (6, 48); the greatest profit is 48 euros, made at a price of 6 euros
    2. (b) T = 22; the temperature falls towards but never goes below 22 °C (the temperature of the kitchen); 94.4 °C
    3. (c) a = 200; N = 50; 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) = 12; n = −5 and C = 4
    5. (e) Exponential; each value is 2 times the one before
    1. (a) f⁻¹(x) = e^(x − 2); domain: all real x; range: f⁻¹(x) > 0
    2. (b) g⁻¹(x) = 3 + √(x − 1); domain x ≥ 1; range g⁻¹(x) ≥ 3
    3. (c) x = 0.159 or x = 3.15

jmaths.xyz · AI HL · 12.5.10 Unit review