Two functions are shown below.

f (x) = 1/2 • 2^x

g (x) = 5x + 2

What is the largest integer value of x such that f (x) ≤ g (x) ?

looking for step by step

x = 6

Step-by-step explanation:

As the function f (x) is an exponencial function, it will grow faster than g (x), that is a linear function.

For small values of x, we have that f (x) < g (x). For example:

f (1) = 1/2 * 2 = 1

g (1) = 5*1 + 2 = 7

f (2) = 1/2 * 4 = 2

g (2) = 5*2 + 4 = 14

So we just need to check some integer values and see when f (x) will be bigger than g (x). It will not be a big value, as the exponencial function grows very fast.

For x = 5, we have:

f (5) = 1/2 * 32 = 16

g (5) = 5*5 + 4 = 29

For x = 6, we have:

f (6) = 1/2 * 64 = 32

g (6) = 5*6 + 4 = 34

For x = 7, we have:

f (7) = 1/2 * 128 = 64

g (7) = 5*7 + 4 = 39

So the largest integer value of x for f (x) ≤ g (x) is x = 6.

Another way to solve this is by plotting both equations, and then checking where they cross, that is, where f (x) = g (x).