Question

Referring to the graph of $g\left(x\right)$, if $h\left(x\right)=-g\left(x\right)+1$ then which of the following statements is/are true?

• A. The range of $h\left(x\right)$ is $-3\le y\le 3$
• B. The minimum value of $h\left(x\right)$ is $-4$
• C. The coordinates of point $A$ on the function $h\left(x\right)$ are $(2,4)$
• D. The graph of $h\left(x\right)$ is increasing between $x=-5$ and $x=-2$

Answer 1

Answer: (A), (D)

The correct options are
A The range of $h\left(x\right)$ is $-3\le y\le 3$
D The graph of $h\left(x\right)$ is increasing between $x=-5$ and $x=-2$

The graph of $h\left(x\right)=-g\left(x\right)+1$ is a vertical reflection of $g\left(x\right)$, over the $x$-axis, that is then shifted up $1$ unit. The graph looks like the dashed line in the following graph as shown in the figure.
Next check with the options: First take the option A
The range of $h\left(x\right)$ is the set of $y$-values from lowest to highest (based on the dashed line). The lowest point occurs at point B and has a $y$-value of $-3$.
The highest value occurs at both ends of the graph and is $3$, so the range is $-3\le y\le 3$.
Therefore, option A is correct.