Question

unit test
unit test complete

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?

the range is the same for both functions: {y | y is a real number}.
the range is the same for both functions: {y | y ≥ 0}.
the range changes from {y | y ≥ 0} to {y | y ≥ 2}.
the range changes from {y | y ≥ 0} to {y | y ≥ 6}.

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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 the range is $\{ y \mid y \geq 0 \}$.

Step2: Determine the new function

Translating 6 units right gives $|x - 6|$, then 2 units up gives $g(x) = |x - 6| + 2$.

Step3: Find range of new function

$|x - 6| \geq 0$, so $|x - 6| + 2 \geq 2$. The range is $\{ y \mid y \geq 2 \}$.

Step4: Compare ranges

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

Answer:

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