Question

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

Hint:

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) = 2^{x}

Let x = 0 - h(x) = 2^{0} = 1

Let x = 2 - h(x) = 2^{2} = 4

Let x = 7- h(x) = 2^{7} = 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) = 2x^{2} + 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.

^{x}

Let x = 0 - h(x) = 2

^{0}= 1

Let x = 2 - h(x) = 2

^{2}= 4

Let x = 7- h(x) = 2

^{7}= 128

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) = 2x

^{2}+ 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.