Functions: Solved Examples

Now let us solve a few examples of functions to understand functions in maths.

Q1. Function f is defined by f(x) = - 3 x 2 + 4 x – 13.Find f(3).

Answer & Explanation: f(3) = - 3 (3)² + 4 (3) - 13 f(3) = -27 + 12 -13 f(3) = -28

Q2. Function h is defined by h(x) = 7x³ + 5x – 2.Find h(-2)

Answer & Explanation: h(-2) = 7(-2)³ + 5(-2) – 2 h(-2) = 7(-8) -10 -2 h(-2) = -68

Q3.If h(2,3) = 17 & h(5,4) = 141.Find h(3,4)

Answer & Explanation: The function h(a,b) here represents a³ + b² h(2,3) = 2³ + 3² = 8+ 9 = 17 h(5,4) = 5³ + 4² = 125 + 16 =141 Therefore, h(3,4) = 3³ + 4² = 27 + 16 = 43

Q4.Defined that x # y = x³ + y² – 3xy, then 4 # (5 # 6) = ?

Answer & Explanation: 4 # (5 # 6) = 4 # (125 + 16 – 90) = 4 # (51) 4 # 51 = 64 + 2601 – 612 = 2053

Q5.Defined that a_{k + 2} = 2a_{k + 1} + a_k + a_{k - 1}. If a₀=2, a₁=3, a₂=4, then a₅=

Answer & Explanation: a_{k + 2} = 2a_{k + 1} + a_k + a_{k - 1} Put the value of k = 1, a_{1 + 2} = 2a_{1 + 1} + a₁ + a_{1 - 1} Therefore, a₃ = 2a₂ + a₁ + a₀ Substituting values a3 = 2 (4) + 3 + 2 = 13 Using the same approach, a₄ = 33 & a₅ = 83.

Q6.Given that, f(x) = x² + 1 ; if x is even if(x) = 3x + 2; if x is odd. Find f(f(2))

Answer & Explanation: f(f(2)) =f (2² + 1) = f (5) Now f(5) = 3(5) + 2 = 17

Q7.Given f(x) = 2x-5, h(x) = x3 + 3, g(x) = x2 + 3. Find f(h(g(2))).

Answer & Explanation: f(h(g(2))) = f(h(22 + 3) = f(h(7)) f(73 + 3) = f(346) = 2(346) – 5 = 687

Q8.Given f(y) = y³ + 7, g(y) = 2y + 3, h(y) = y – 4. Find f(2) + g(2) – h(2)

Answer & Explanation: f(2) = 2³ + 7 = 15 g(2) = 2(2) + 3 = 7 h(2) = 2 – 4 = -2 Now, f(2) + g(2) – h(2) = 15 + 7 – (-2) = 24

Q9.If k₁ = 1 and k_{n + 1} – 3k_n + 2 = 4n, for every positive integer n, Find k₂₅.

Answer & Explanation: Now, k₁ = 1, K₂ – 3k₁ + 2 = 4n K₂ – 3(1) + 2 = 4(1) K₂ = 5 By solving this equation, we get k₂ = 5 Similarly, k₃ = 21 & k₄ = 73. Looking at these values we can generalise, that k_n = 3ⁿ – 2n Therefore, k₂₅ = 3²⁵ – 50

Q10.Function f is defined by f(x) = x² - 3 ; if x is even. Function f is defined by f(x) = x² + 1 ; if x is odd. Find f(f(2).

Answer & Explanation: f(f(2)) = f(2² - 3) = f (4-3) = f(1) F(1) = 1² + 1 = 1 + 1 = 2