Question

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}.

Explanation:

Step1: Analyze original function range

The function \( f(x) = |x| \) has a range of \( \{ y | y \geq 0 \} \) because the absolute value of any real number is non - negative.

Step2: Analyze translation effects on range

When a function is translated horizontally (right or left), it does not affect the range (the set of y - values). When a function is translated vertically, a vertical shift up by \( k \) units will shift the range up by \( k \) units. Here, the function \( f(x)=|x| \) is translated 6 units to the right (horizontal translation, no effect on range) and 2 units up (vertical translation). The original range is \( y\geq0 \), after shifting up 2 units, the new range is \( y\geq0 + 2=2 \), so the range changes from \( \{ y|y\geq0\} \) to \( \{ y|y\geq2\} \).

Answer:

The range changes from \(\{ y\mid y\geq0\}\) to \(\{ y\mid y\geq2\}\).