Consider the function $f(x)=-2x-1$ with the domain restricted to the interval $D(f)=[ -3, 4)$. Peter was tasked with finding its range.
He proceeded as follows:
(1) First, he substituted the left endpoint of the domain into the function’s equation: $$ f(-3)=-2(-3)-1=6-1=5 $$
(2) Next, he noticed that the given domain is open on the right, meaning it does not include the right endpoint, $x=4$. Peter then reasoned as follows: “I will consider the linear function with the same equation, $g(x)=-2x-1$, but with its domain extended to $\mathbb{R}$. If I substitute $x= 4$ into this equation, I get:” $$ g(4)=-2\cdot 4-1=-8-1=-9 $$
(3) Finally, Peter thought: “Since this is a linear function, it is monotonic. Therefore, the previous information is enough to determine the range. The function value at the left endpoint is $5$, and as $x$ approaches the right endpoint, the function value approaches $-9$. However, since the right endpoint is not included in the domain, $-9$ is not part of the range. Thus, the final result is:” $$ H(f)=[ 5,-9) $$
Did Peter make any mistakes in his procedure? If so, identify, where.
Yes, he made a mistake in step (3). He correctly proceeded in steps (1) and (2), however he incorrectly determined the range. It should be $H(f)=(-9,5] $ because the left endpoint of an interval must always be smaller than the right one.
Yes, he made a mistake in step (3). He correctly proceeded in steps (1) and (2), however he incorrectly determined the range. It should be $H(f)=[ -9, 5)$, because the domain was given as a closed interval on the left.
Yes, he made a mistake in step (2). When determining the range, we cannot substitute the endpoints into a different function (a function with a different domain).
Yes, he made a mistake in step (1). The correct calculation is: $$ f(-3)=-6-1=-7 $$
No, all steps are correct.
The picture shows the graph of the given function $f$. From the graph, it is clearly observed that the range of $f$ is: $$ H(f)=(-9,5] $$
