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Functions Test - 4

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Functions Test - 4
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  • Question 1
    3 / -1

     Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) = 3x + 4. Then the composition of f and g is ____________

    Solution

    The composition of f and g is given by f(g(x)) which is equal to 2(3x + 4) + 1.

  • Question 2
    3 / -1

    A certain function always obeys the rule: If f (x.y) = f(x). f(y) where x and y are positive realnumbers. A certain Mr. Mogambo found that the value of f (128) = 4, then find the value of thevariable M = f (0.5). f (1). f (2). f (4). f (8). f (16). f (32). f (64). f (128). f (256)

    Solution

    Since f (128) = 4, we can see that the product of f (256). f (0.5) = f (256 × 0.5) = f (128) = 4.
    Similarly, the products f (1). f (128) = f (2). f (64)
    = f (4). f (32) = f (8). f (16) = 4.
    Thus, M = 4 × 4 × 4 × 4 × 4 = 1024.
    Option (d) is the correct answer.

  • Question 3
    3 / -1

    If f(t) = 2t, then f(0), f(1), f(2) are in

    Solution

    f(0) = 1, f(1) = 2 and f(2) = 4
    Hence, they are in G.P.

  • Question 4
    3 / -1

    The graph of y = (x + 3)3 + 1 is the graph of y = x3 shifted

    Solution

    (x + 3)3 would be shifted 3 units to the left and hence (x + 3)3 + 1 would shift 3 units to the left
    and 1 unit up. Option (c) is correct.

  • Question 5
    3 / -1

    If f(x) = x2 and g(x) = logex, then f(x) + g(x) will be

    Solution

    (x2 + loge x) would be neither odd nor even since it obeys neither of the rules for even function
    (f(x) = f(–x)) nor for odd functions (f(x) = –f(–x)).

  • Question 6
    3 / -1

    f(x) is any function and f–1(x) is known as inverse of f(x), then f–1(x) of f(x) = ex is

    Solution

    y = ex
    fi loge y = x.
    fi f –1(x) = loge x.

  • Question 7
    3 / -1

    Which of the following functions will have a minimum value at x = –3?

    Solution

    If you differentiate each function with respect to x, and equate it to 0 you would see that for none
    of the three options will get you a value of x = –3 as its solution. Thus, option (d) viz. None of
    these is correct.

  • Question 8
    3 / -1

    Define the following functions:

    f(x, y, z) = xy + yz + zx

    g(x, y, z) = x2y + y2z + z2x and

    h(x, y, z) = 3 xyz

    Q.

    Find the value of the following expressions:37. h[f(2, 3, 1), g(3, 4, 2), h(1/3, 1/3, 3)]

    Solution

    The given function would become h[ 11, 80, 1] = 2640.

  • Question 9
    3 / -1

    Define the following functions:
    f(x, y, z) = xy + yz + zx
    g(x, y, z) = x2y + y2z + z2x and
    h(x, y, z) = 3 xyz
    Find the value of the following expressions:

    Q.

    f[ f (1, 1, 1), g(1, 1, 1), h(1, 1, 1)]

    Solution

    The given function would become f[3, 3, 3] = 27.

  • Question 10
    3 / -1

    If f(x) = 1/ g(x), then which of the following is correct?

    Solution

    The number of g’s and f’s should be equal on the LHS and RHS since both these functions are
    essentially inverse of each other.
    Option (c) is correct.

  • Question 11
    3 / -1

    If R(a/b) = Remainder when a is divided by b;
    Q(a/b) = Quotient obtained when a is divided by b;
    SQ(a) = Smallest integer just bigger than square root of a.

    Q.

    If a = 12, b = 5, then find the value of SQ[R {(a + b)/b}].

    Solution

    SQ [R[(a + b)/b]] = SQ [R[17/5]] fi SQ [2] = 2.

  • Question 12
    3 / -1

    If R(a/b) = Remainder when a is divided by b;
    Q(a/b) = Quotient obtained when a is divided by b;
    SQ(a) = Smallest integer just bigger than square root of a.

    Q.

    If a =18, b = 2 and c = 7, then find the value of Q [{SQ(ab) + R(a/c)}/b].

    Solution

    Q [[SA (36) + R (18/7)]/2] = Q [(7 + 4)/2] = Q [11/2] = 5.

  • Question 13
    3 / -1

    Read the following passage and try to answer questions based on
    them.
    [x] = Greatest integer less than or equal to x
    {x} = Smallest integer greater than or equal to x.

    Q.

    If x is not an integer, then ({x} + [x]) is

    Solution

    [x] + {x} will always be odd as the values are consecutive integers.

  • Question 14
    3 / -1

    If f(t) = t2 + 2 and g(t) = (1/t) + 2, then for t = 2, f [g(t)] – g[f(t)] = ?

    Solution

    f(g(t)) – g(f(t)) = f(2.5) – g(6) = 8.25 – 2.166 = 6.0833.

  • Question 15
    3 / -1

    Let F(x) be a function such that F(x) F(x + 1) = – F(x – 1)F(x–2)F(x–3)F(x–4) for all x ≥ 0.Given the values of If F (83) = 81 and F(77) = 9, then the value of F(81) equals to

    Solution

    When the value of x = 81 and 82 is substituted in the given expression, we get,
    F (81) F (82) = – F (80) F (79) F(78) F(77)
    F (82) F (83) = – F (81) F (80) F(79) F(78)
    On dividing (i) by (a), we get

    Option (a) is the correct answer.

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