the graphs $f(x)=4x$ and $g(x)=4^{x}$ intersect at $x = \frac{1}{2}$ and $x = 1$. select all of the…

the graphs $f(x)=4x$ and $g(x)=4^{x}$ intersect at $x = \frac{1}{2}$ and $x = 1$. select all of the intervals of $x$ for which $g(x)>f(x)$. a. $(-\\infty,\\infty)$ b. $(-\\infty,\frac{1}{2})$ c. $(-\\infty,1)$ d. $(\frac{1}{2},1)$ e. $(\frac{1}{2},\\infty)$ f. $(1,\\infty)$

the graphs $f(x)=4x$ and $g(x)=4^{x}$ intersect at $x = \frac{1}{2}$ and $x = 1$. select all of the intervals of $x$ for which $g(x)>f(x)$. a. $(-\\infty,\\infty)$ b. $(-\\infty,\frac{1}{2})$ c. $(-\\infty,1)$ d. $(\frac{1}{2},1)$ e. $(\frac{1}{2},\\infty)$ f. $(1,\\infty)$

Answer

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 can test values in the intervals separated by these intersection - points.

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

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

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

Let's take $x=\frac{3}{4}$. 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 $2.83 < 3$, $g(x)<f(x)$ for $x\in(\frac{1}{2},1)$.

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

Let's take $x = 2$. Then $f(2)=4\times2 = 8$ and $g(2)=4^{2}=16$. Since $16>8$, $g(x)>f(x)$ for $x\in(1,\infty)$.

Answer:

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