Question

the function $f(x)$ is graphed below. write the domain and range using inequality notation.
$f(x) = -2|x - 3| + 8$
draw
show your work here
domain:
range:

Explanation:

Step1: Determine the domain

The function is an absolute - value function of the form \(y = a|x - h|+k\). For any absolute - value function, there are no restrictions on the values of \(x\) that we can input (we can plug in any real number for \(x\)). Looking at the graph, the line extends infinitely to the left and to the right. So, the domain of the function \(f(x)=- 2|x - 3|+8\) is all real numbers. In inequality notation, this is \(-\infty

Step2: Determine the range

The general form of an absolute - value function is \(y=a|x - h|+k\). The vertex of the absolute - value function \(f(x)=-2|x - 3|+8\) is at \((h,k)=(3,8)\). The coefficient \(a=-2\) is negative, which means the graph of the absolute - value function opens downwards. For an absolute - value function that opens downwards, the maximum value of the function occurs at the vertex. The \(y\) - value of the vertex is \(y = 8\), and as \(x\) moves away from \(x = 3\) (both to the left and to the right), the value of \(y\) decreases without bound (since the graph extends downwards infinitely). So, the range of the function is all real numbers less than or equal to \(8\). In inequality notation, this is \(y\leq8\) (or \(-\infty

Answer:

Domain: \(-\infty < x<\infty\)
Range: \(y\leq8\) (or \(-\infty < y\leq8\))