Question

What is the smallest positive integer for x so that f(x)=200(2)* is greater than the value of g(x)=500x+400?

Answer 1

We have to functions, namely:

$f(x)=200(2)^{x} \ and \ g(x)=500x+400$

So the problem is asking for the smallest positive integer for $x$ so that $f(x)$ is greater than the value of $g(x)$, that is:

$f(x)\ \textgreater \ g(x) \\ \therefore 200(2)^{x}\ \textgreater \ 500x+400$

Let's solve this problem by using the trial and error method:

$for \ x=1 \\f(1)=400 \\ g(1)=900 \\ Then \ f(1) \ \textless \ g(1) \\ \\ \\ for \ x=2 \\f(2)=800 \\ g(2)=1400\\ Then \ f(2)\ \textless \ g(2) \\ \\ \\ for \ x=3 \\f(3)=1600 \\ g(3)=1900 \\ Then \ f(3)\ \textless \ g(3) \\ \\ \\ for \ x=4 \\f(4)=3200 \\ g(4)=2400 \\ \boxed{Then \ f(4)\ \textgreater \ g(4)}$

So starting $x$ from 1 and increasing it in steps of one we find that:

$f(x)>g(x)$

when $x=4$

That is, the smallest positive integer for $x$ so that the function $f(x)$ is greater than $g(x)$ is 4.