the graph of $f(x) = |x|$ is stretched by a factor of 0.3 and translated down 4 units. which statement about…

Brief Explanations:

1. First, identify the parent function $f(x)=|x|$:
   • Domain of $f(x)=|x|$: All real numbers (you can input any real $x$ into the absolute value function).
   • Range of $f(x)=|x|$: All real numbers greater than or equal to 0 ($y\geq0$), since absolute value is never negative.
2. Analyze the transformed function: A vertical stretch by 0.3 and translation down 4 units gives $g(x)=0.3|x|-4$.
   • Domain of $g(x)$: Vertical stretches and vertical translations do not restrict the input values, so the domain remains all real numbers, matching the parent function.
   • Range of $g(x)$: The parent range $y\geq0$ stretched by 0.3 is still $y\geq0$, then shifting down 4 units gives $y\geq-4$, which is different from the parent range.
3. Evaluate each option:
   • Option 1: Incorrect, the ranges are not $\geq4$.
   • Option 2: Incorrect, the domains are the same.
   • Option 3: Incorrect, the transformed range is $\geq-4$, not $\geq0$.
   • Option 4: Correct, both functions have a domain of all real numbers.

Answer:

The domain of the transformed function and the parent function are both all real numbers.