the graph of $f(x) = |x|$ is translated 6 units to the right and 2 units up to form a new function. which…

the graph of $f(x) = |x|$ is translated 6 units to the right and 2 units up to form a new function. which statement about the range of both functions is true?\nthe range is the same for both functions: ${y \\mid y \\text{ is a real number}}$.\nthe range is the same for both functions: ${y \\mid y \\geq 0}$.\nthe range changes from ${y \\mid y \\geq 0}$ to ${y \\mid y \\geq 2}$.\nthe range changes from ${y \\mid y \\geq 0}$ to ${y \\mid y \\geq 6}$.

the graph of $f(x) = |x|$ is translated 6 units to the right and 2 units up to form a new function. which statement about the range of both functions is true?\nthe range is the same for both functions: ${y \\mid y \\text{ is a real number}}$.\nthe range is the same for both functions: ${y \\mid y \\geq 0}$.\nthe range changes from ${y \\mid y \\geq 0}$ to ${y \\mid y \\geq 2}$.\nthe range changes from ${y \\mid y \\geq 0}$ to ${y \\mid y \\geq 6}$.

Answer

Explanation:

Step1: Find range of original function

The original function is $f(x)=|x|$. The absolute value of any real number is non-negative, so its range is ${y \mid y \geq 0}$.

Step2: Define translated function

A translation 6 units right and 2 units up transforms $f(x)$ to $g(x)=|x-6|+2$.

Step3: Find range of new function

The term $|x-6|$ is non-negative ($\geq0$). Adding 2 to this gives $|x-6|+2 \geq 0+2=2$. So the range of $g(x)$ is ${y \mid y \geq 2}$.

Step4: Compare the two ranges

The original range is ${y \mid y \geq 0}$, and the new range is ${y \mid y \geq 2}$.

Answer:

The range changes from ${y \mid y \geq 0}$ to ${y \mid y \geq 2}$.