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Easy

Question

Compare the functions f(x)= 3X+2 , g(x)= 2x2+3 and h(x)= 2x . Show that as x increases , h(x) will eventually exceed f(x) and g(x).

hintHint:


We observe that-
f(x) is a linear function (highest power is 1),
g(x) is a quadratic function (highest power is 2) and
h(x) is an exponential function (power is a variable).
We will simply substitute different values of x in the given functions and plot the same on a graph and compare the observations.

The correct answer is: Hence, proved that f(x) is a Linear function, g(x) is a quadratic function & h(x) is an exponential function, where h(x) exceeds f(x) and g(x) in the long run.


    Step-by-step solution:-

    h(x) = 2x
    Let x = 0 - h(x) = 20 = 1
    Let x = 2 - h(x) = 22 = 4
    Let x = 7- h(x) = 27 = 128

    ∴ We plot the points (0,1); (2,4) & (7,128) for h(x)
    f(x) = 3x + 2
                                                                                  Let x = 0 - f(x) = 3(0) + 2 = 0 + 2 = 2
                                                                                  Let x = 2 - f(x) = 3(2) + 2 = 6 + 2 = 8
                                                                                Let x = 7- f(x) = 3(7) + 2 = 21 + 2 = 23
    ∴ We plot the points (0,2); (2,8) & (7,23) for f(x)
    g(x) = 2x2 + 3
                                                                                 Let x = 0 - g(x) = 2(0)2 + 3 = 0 + 3 = 3
                                                                                Let x = 2 - g(x) = 2(2)2 + 3 = 8 + 3 = 11
                                                                               Let x = 7- g(x) = 2(7)2 + 3 = 98 + 3 = 101
    ∴ We plot the points (0,3); (2,11) & (7,101) for g(x)
    From the adjacent graph, we observe that-
    Line representing f(x) is a straight line. Hence, f(x) is a linear function.
    Line representing g(x) & h(x) are not a straight line. Hence, these are polynomial functions.
    However, as increase the x variable, h(x) being an exponential function, increases faster than g(x).
    Hence, we see that h(x) will eventually exceed f(x) and g(x).
    Final Answer:-
    ∴ Hence, proved that f(x) is a Linear function, g(x) is a quadratic function & h(x) is an exponential function, where h(x) exceeds f(x) and g(x) in the long run.

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