Answer:
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).
Answer:
x = 6
Step-by-step explanation:
We assume your functions are ...
In general, mixed polynomial and exponential functions have no algebraic solution. That means we need to solve f(x) = g(x) using a table of values, or by graphing.
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Attached is a graphical solution. It shows the largest value of x for which f(x) = g(x) is x=6.
Check: f(6) = 1/2(2^6) = 32 = g(6) = 5(6) +2
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Additional comment
We can start with f(x) = g(x) and subtract g(x) from both sides of the equation. This gives ...
f(x) -g(x) = 0
Many graphing calculators readily identify the zeros of a function or expression. We have used that fact to find the two values of x where these functions are equal: ≈0.32 and 6.
For x>6, the exponential function increases without bound, so the graphs never cross again. Likewise, for x < -0.32, the linear function decreases without bound, so the graphs never cross again.
The lower value of x where the graphs cross is an irrational number near −0.319886722135.
