Let us solve a few practice problems of Functions to understand the concept of Functions in math better.

Q.1. If f(x) = 3x + 2 & g(x) = x

^{2} – 1. Find f(g(-3))

Answer & Explanation

Sol : Option A
Explanation: f(g(-3)) = f((-3)^{2} -1) = f(8) = 3(8) + 2 = 26

Q.2. Given f(x) = x

^{2} + x ; if x is a prime number.f(x) = 2x + 5 ; if x is non- prime

Find f(f(1))

Answer & Explanation

Sol : Option C
Explanation: Now, 1 is neither prime nor composite So f(2(1) + 5) = f(7) = 7^{2} + 7 = 56

Q.3. Given f(x) = x3 + 1,

g(x) = 2x – 5

h(x) = [f(x)]2 – g(x)

Find h(-2)

a) 34

b) -23

c) -12

d) 58

Answer & Explanation

Sol : Option D
Explanation: h(-2) = [f(-2)]^{2} – g(-2) = [(-2)^{3} + 1]^{2} – [2(-2) -5] [-8 + 1]^{2} – [-4-5] = [-7]^{2} – [-9] = 49 + 9 = 58

Q.4. Given g(x) = x3 – x2 + 2 ; if x is an odd number

g(x) = 2x + 5 ; if x is an even number

Find g(g(3))

Answer & Explanation

Sol : Option D
Explanation: Now, 3 is an odd number So, g(3^{3} – 3^{2} + 2) = g(27 – 9 + 2) = g(20) = 2(20) + 5 = 45

Q.5. f(x,y) = x + y ; if both x & y are even

f(x,y) = x – y ; if both x & y are odd

f(x, y) = 2x – y

^{2} , if any one of them is odd & other is even.

Find f(2, 3) – f(2, 4)

a) -31

b) -56

c) -11

d) -13

Answer & Explanation

Sol : Option C
Explanation: f(2, 3) = 2(2) – 3^{2} = 4 – 9 = -5 f(2, 4) = 2 + 4 = 6 Now, f(2, 3) – f(2, 4) = -5 – 6 = -11

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Q.6. f(x,y) = x

^{2} + y ; if both x & y are prime

f(x,y) = x + 2y ; if both x & y are composite

f(x, y) = x + y , if any one of them is prime & other is composite.

Find f(5, 3) + f(3, 6)

Answer & Explanation

Sol : Option B
Explanation: f(5, 3) = 5^{2} + 3 = 25 + 3 = 28 F(3, 6) = 3 + 6 = 9 Now, f(5, 3) + f(3, 6) = 28 + 9 = 37

Directions for questions 7 to 10: Read the instructions and solve the questions accordingly

x # y = x + y
x @ y = x – y
x & y = x^{2} – y^{2}
x $ y = (x + y)^{2}

Q.7. Find 3 # (4 $ 6)

a) 23

b) 256

c) 103

d) 109

Answer & Explanation

Sol : Option C
Explanation: 3 # (4+6)^{2} = 3 # 100 = 3 + 100 = 103

Q8. Find [3 @ (7 # ( 5 & 3) ]

a) -20

b) -23

c) -34

d) -45

Answer & Explanation

Sol : Option A
Explanation: [ 3 @ (7 # (5^{2} – 3^{2} )] = [ 3 @ (7 # 16)] [3 @ (7 + 16)] = [ 3 @ 23] = [ 3 – 23] = -20

Q9. Find -3 $ (6 & (4 # 3))

a) 361

b) 121

c) 144

d) 987

Answer & Explanation

Sol : Option A
Explanation: -3 $ (6 & (4 + 3)) = -3 $ (6 & 7) -3 $ (6^{2} – 7^{2} ) = -3 $ (36 – 49) = -3 $ -13 = (-3-16)^{2} = (-19)^{2} = 361

Q10. . Find the 3 $ (4 & (5 & 6))

a) -102

b) -56

c) -34

d) -23

Answer & Explanation

Sol : Option A
Explanation: 3 $ (4 & (5^{2} – 6^{2} )) = 3 $ (4 & (25 – 36)) 3 $ (4 & -11) = 3 $ (16 -121) = 3 $ -105 = 3 -105 = -102