First, sketch a graph of the piecewise function𝑓(𝑥)={2𝑥+−7𝑥2−6𝑥<1𝑥≥1.Use your sketch to answer the following questions.(a) lim𝑥→1−𝑓(𝑥)= (b) lim𝑥→1+𝑓(𝑥)= (c) 𝑓(1)=

To answer this question, we first need to understand the piecewise function and how to sketch it.

The function f(x) is defined as follows:

• For x < 1, f(x) = 2x
• For x ≥ 1, f(x) = -7x^2 - 6x

Now, let's sketch the graph:

1. For x < 1, the graph is a straight line with a slope of 2. It passes through the origin (0,0) and continues until x = 1 (but does not include x = 1).

2. For x ≥ 1, the graph is a downward-opening parabola with a vertex at x = -b/2a = 6/14 = 0.43 (approximately). This part of the graph starts at x = 1 (including x = 1).

Now, let's answer the questions:

(a) The limit as x approaches 1 from the left (lim x→1− f(x)) is the y-value of the function as x gets infinitely close to 1 from values less than 1. Looking at the graph, this would be the y-value of the straight line at x = 1, which is 2*1 = 2.

(b) The limit as x approaches 1 from the right (lim x→1+ f(x)) is the y-value of the function as x gets infinitely close to 1 from values greater than 1. Looking at the graph, this would be the y-value of the parabola at x = 1, which is -71^2 - 61 = -13.

(c) The value of f(1) is the y-value of the function at x = 1. Since x = 1 falls in the second part of the piecewise function (x ≥ 1), we use the equation for the parabola to find f(1) = -71^2 - 61 = -13.

So, the answers are: (a) 2 (b) -13 (c) -13