Question

Write the equation of the piecewise function that is represented by its graph.

Answer 1

From the figure, the green color graph represent the rational function and red color graph represent the line.

From the figure, the domain of the rational function graph (- ∞, 0) U (0, 1], and the domain of the line graph [1, ∞).

 

Find the rational function :

The general form of rational function f(x) = N(x) / D(x), where N(x) and D(x) are polynomial functions. The line x = 0 is a vertical asymptote of the graph of f(x) as shown in Figure. From this figure, you can see that the graph of f(x) also has a horizontal asymptote—the line y = 0. This means that the values of f(x) = 1/x approach zero as x increases or decreases without bound.

 

Find the line equation :

From the figure, the red line passes through the points (1,1) and (8, 2).

The points are (x₁, y₁) = (1, 1) and (x₂, y₂) = (8, 2).

To find the line equation by using two points form of line equation is y - y₁ = [ (y₂ - y₁) / (x₂ - x₁) ] * (x - x₁).

y - 1 = [ (2 - 1) / (8 - 1) ] * (x - 1)

y - 1 = (1/7)*(x - 1)

y - 1 = x/7 - 1/7

y = x/7 - 1/7 + 1

y = x/7 + 6/7.

The line equation is f(x) = (x + 6)/7 and rational function f(x) = 1/x.

Write the piecewise function by using these two functions f(x) = 1/x and f(x) = (x + 6)/7 with their domains.

The piecewise function