Select the correct graph of (f(x)) and state the d

Questions: Select the correct graph of (f(x)) and state the domain and range. (f(x)=leftbeginarrayll -x, x<-2 6, -2<x<2 x^2, x geq 2 endarrayright.)

Transcript text: Select the correct graph of $f(x)$ and state the domain and range. \[ f(x)=\left\{\begin{array}{ll} -x, & x<-2 \\ 6, & -2

Solution

Solution Steps

Step 1: Analyze the piecewise function

The function _f(x)_ is defined as follows:

_f(x) = -x_ when _x < -2_ This is a straight line with a slope of -1. For x values less than -2, the graph will have a positive output value equal to the negative value of x.
_f(x) = 6_ when _-2 ≤ x < 2_ This is a horizontal line at _y = 6_. For x values greater than or equal to -2 and less than 2, the output of f(x) is a constant 6.
_f(x) = x²_ when _x ≥ 2_ This is a parabola opening upwards. For x values greater than or equal to 2, the graph follows x squared.

Step 2: Identify the correct graph

Graph B correctly represents the piecewise function. The line defined for _x < -2_ is correctly plotted as _f(x) = -x_ (note the open circle at x=-2 since that value is defined differently). The horizontal line _f(x) = 6_ is drawn for _-2 ≤ x < 2_ with a closed circle at x=-2 and an open circle at x=2. Lastly, for _x ≥ 2_, we have _f(x) = x²_ beginning with a closed circle.

Step 3: Determine the domain

The function is defined for all real values of _x_, so the domain is _(-∞, ∞)_.

Step 4: Determine the range

Examining the graph and considering how the three segments define it, we see: For x < -2, the values start from values greater than 2 and then approach infinity. For -2 ≤ x < 2, the graph is a horizontal line at y = 6. For x ≥ 2, the parabolic section outputs y values starting at 4 and going toward infinity.

Therefore, the range starts at 4 then continues until infinity. It does not include the interval between 2 and 4 and then goes from 4 up to infinity. It also has the constant value of 6. As such, the range is _[4, ∞)_.