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

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) = 2^x
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.