Question

In Exercises 11 and 12, determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g .\) (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=x+1, \quad g(x)=(x-1)^{2}\) (a) -2 (b) 2 (c) 10 (d) 4 (e) 8

Short answer

(b) 2

Step-by-step solution

Sketch the Graph

Begin by sketching the graphs of the two functions. The graph of \(f(x) = x + 1\) is a straight line with slope 1 and y-intercept 1. The graph of \(g(x) = (x-1)^2\) is a parabola that opens upwards and is shifted 1 unit to the right. Plot these functions together on the same set of axes.

Find the Intersection Points

Find the points where the graphs of the functions intersect each other. Observing the graph, we can see that the functions intersect at two points, \(x = -1\) and \(x = 2\). These are the points where \(f(x)\) and \(g(x)\) are equal (i.e., when \(x+1 = (x - 1)^2\) ).

Estimate the Area

Estimate the area enclosed by the two functions between the intersection points you identified. The area enclosed by the two functions appears to be slightly more than the area of the three triangles that are formed by the x-axis and lines x=-1, x=1 and x=2 where their heights are each less than 2 (from visual inspection). So the area is less than 3 half-rectangles of 2 by 2, but more than 2 such triangles. Hence it is expected to be slightly more than 2, but definitely less than 4.

Key concepts

Sketching Graphs

Understanding the concept of sketching graphs is essential to visualize and solve many mathematical problems. To sketch the graph of a linear function like f(x) = x + 1, start by identifying the y-intercept, which is the point where the graph crosses the y-axis. In this case, the y-intercept is at (0,1). The slope of the line is the coefficient of x, which tells us that for each increase of 1 in x, y will increase by 1 as well. Draw this line on a set of axes.

For the quadratic function g(x) = (x-1)^2, plot the vertex of the parabola, which is at (1,0) since the function is shifted 1 unit to the right from a standard y = x^2 parabola. The parabola opens upwards, and its symmetry can be used to plot additional points and sketch the curve. It's always helpful to find a few more points by substituting values of x into the function and plotting the resulting points on the graph.

Once both graphs are drawn on the same set of axes, the visual representation will facilitate the understanding of their relationship and the regions they define.

Intersection of Functions

Finding the intersection of functions involves determining where two graphs coincide. This process starts with setting the two functions equal to each other and solving for the variable, which corresponds to the x-coordinate(s) of the point(s) where both graphs meet. For example, to find where f(x) = x + 1 and g(x) = (x - 1)^2 intersect, set x + 1 = (x - 1)^2 and solve for x.

In graphical terms, the intersection points are where the plotted lines or curves touch or cross each other. In our case, after solving the equation, we've found that x = -1 and x = 2 are the intersection points, which can also be verified on the graph. It is crucial to locate these points accurately, as they often serve as boundaries for the region whose area we are interested in estimating or calculating.

Estimating Area

Estimating area between curves can be thought of as finding the size of a 'patch' enclosed by the graphs of two or more functions. To start, identify the region bounded by the curves between their intersection points. The goal is to approximate how much space this region occupies without necessarily calculating the exact value.

In this exercise, a visual approach is suggested to approximate the area. Imagine fitting geometric shapes such as triangles or rectangles within the bounded region to get a sense of its size. From the sketches, we note that the area of the region enclosed by f(x) and g(x) is more than two 2x2 triangles but less than three of such triangles, leading us to conclude that the area is slightly greater than 4 but less than 6. However, the best available estimate given in the options is 4, which is option (d).

Although it's an approximation, this method gives a solid foundation for understanding the concept of area between curves, which can then be further explored with exact methods such as integration for a more precise result.