Respuesta :

frika

Answer:

C.

[tex]f(x)=\left\{\begin{array}{l}\dfrac{1}{2}x-3,\ \ x<2\\ \\3x-8,\ \ x\ge 2\end{array}\right.[/tex]

Step-by-step explanation:

The graph of the piecewise function consists of two linear functions which graphs meets at point with x-coordinate x = 2.

1. The left linear function is determined for all x < 2 and is passing through the points (0,-3) and (-2,-4). So, its slope is

[tex]\dfrac{-4-(-3)}{-2-0}=\dfrac{-1}{-2}=\dfrac{1}{2}[/tex]

and its equation is

[tex]y-(-3)=\dfrac{1}{2}(x-0)\\ \\y=\dfrac{1}{2}x-3[/tex]

2. The right linear function is determined for all x ≥ 2 and is passing through the points (2,-2) and (3,1). So, its slope is

[tex]\dfrac{1-(-2)}{3-2}=\dfrac{3}{1}=3[/tex]

and its equation is

[tex]y-1=3(x-3)\\ \\y=3x-8[/tex]

3. Hence, the expression for the piece-wise function is

[tex]f(x)=\left\{\begin{array}{l}\dfrac{1}{2}x-3,\ \ x<2\\ \\3x-8,\ \ x\ge 2\end{array}\right.[/tex]

The slope of a line is the rate of change. The correct option is the first option.

What is the slope of a line?

The slope of a line is the rate of change, also it can be described as the gradient of a line is a number that helps to know both the direction and the steepness of the line.

In order to know the correct option of function that describes the graph, we will find the function of the graph, therefore,

Let's find the equation of the first function that is on the left side of the graph.

We know that the slope of the line is given by the formula,

[tex]\text{Slope of the line} = \dfrac{y_2-y_1}{x_2-x_1}[/tex]

Considering the two of the points of the slope,

[tex](x_1, y_1) = (2,-2)\\(x_2, y_2) = (-4,-5)\\[/tex]

Substituting the values to know that slope of the line, therefore,

[tex]\text{Slope of the line}, m= \dfrac{-5-(-2)}{-4-2} = \dfrac{-3}{-6} = \dfrac{1}{2}[/tex]

Now, using one of the points to know the value of the constant,

[tex]y=mx+c\\y_2 = mx_2+c\\-5=(\dfrac{1}{2}\times -4)+c\\-5=-2+c\\-3=c[/tex]

Hence, the equation that will represent the graph for the values of x less than -2 is [tex]y=\dfrac{1}{2}x-3[/tex].

Since, there are only two options that match this condition, the first option and the third option, now other functions will represent the rest of the graph, therefore, for the value of x greater than -2 (x≥-2).

Hence, the correct option is the first option.

Learn more about Slope of Line:

https://brainly.com/question/2514839

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