which best describes the range of the function $f(x) = 2\\left(\\frac{1}{4}\\right)^x$ after it has been…

which best describes the range of the function $f(x) = 2\\left(\\frac{1}{4}\\right)^x$ after it has been reflected over the $y$-axis?\noptions:\n- all real numbers less than 0\n- all real numbers\n- all real numbers greater than 0\n- all real numbers less than or equal to 0

which best describes the range of the function $f(x) = 2\\left(\\frac{1}{4}\\right)^x$ after it has been reflected over the $y$-axis?\noptions:\n- all real numbers less than 0\n- all real numbers\n- all real numbers greater than 0\n- all real numbers less than or equal to 0

Answer

Explanation:

Step1: Reflect function over y-axis

To reflect a function $f(x)$ over the $y$-axis, replace $x$ with $-x$. The new function is $g(x) = 2\left(\frac{1}{4}\right)^{-x}$

Step2: Simplify the reflected function

Use exponent rule $a^{-x} = \left(\frac{1}{a}\right)^x$ or $a^{-x}=(a^{-1})^x$, so $\left(\frac{1}{4}\right)^{-x}=4^x$. $g(x) = 2(4)^x$

Step3: Analyze range of $g(x)$

For any real $x$, $4^x > 0$. Multiply by 2: $2(4^x) > 0$. The function never equals 0, and outputs all positive real numbers.

Answer:

all real numbers greater than 0