the graphs f(x) = 4x and g(x) = 4^x intersect at x = 1/2 and x = 1. select all of the intervals of x for…

Explanation:

Step1: Analyze intersection points

The functions $f(x) = 4x$ and $g(x)=4^{x}$ intersect at $x = \frac{1}{2}$ and $x = 1$. We need to check the relationship between the two - functions in the intervals separated by these intersection points. The intervals are $(-\infty,\frac{1}{2})$, $(\frac{1}{2},1)$ and $(1,\infty)$.

Step2: Test a value in $(-\infty,\frac{1}{2})$

Let's choose $x = 0$. Then $f(0)=4\times0 = 0$ and $g(0)=4^{0}=1$. Since $g(0)>f(0)$, $g(x)>f(x)$ on the interval $(-\infty,\frac{1}{2})$.

Step3: Test a value in $(\frac{1}{2},1)$

Let's choose $x=\frac{3}{4}$. Then $f(\frac{3}{4})=4\times\frac{3}{4}=3$ and $g(\frac{3}{4}) = 4^{\frac{3}{4}}=\sqrt[4]{4^{3}}=\sqrt[4]{64}\approx2.83$. Since $f(\frac{3}{4})>g(\frac{3}{4})$, $g(x)<f(x)$ on the interval $(\frac{1}{2},1)$.

Step4: Test a value in $(1,\infty)$

Let's choose $x = 2$. Then $f(2)=4\times2 = 8$ and $g(2)=4^{2}=16$. Since $g(2)>f(2)$, $g(x)>f(x)$ on the interval $(1,\infty)$.

Answer:

B. $(-\infty,\frac{1}{2})$, F. $(1,\infty)$