Question

the function ( f(x) = \frac{x^2 - 64}{x - 8} ) is a rational function.

a. determine the coordinates of any removable discontinuities.

b. sketch the graph.

a. select the correct choice and, if necessary, fill in the answer box to complete your choice.

a. the coordinates of all removable discontinuities is/are (\boxed{quad}) (type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)
b. there are no removable discontinuities.

b. choose the graph.

options for graphs: a, b, c, d, e, f, g, h (with corresponding grid graphs shown)

Explanation:

Step1: Factor the numerator

The numerator \(x^2 - 64\) is a difference of squares, so it factors as \((x - 8)(x + 8)\). So the function becomes \(f(x)=\frac{(x - 8)(x + 8)}{x - 8}\) (for \(x
eq8\)).

Step2: Simplify the function

Cancel out the common factor \((x - 8)\) (for \(x
eq8\)), so \(f(x)=x + 8\) (with a hole at \(x = 8\) because the original function is undefined there, but the simplified function is defined there).

Step3: Find the coordinates of the removable discontinuity

To find the \(y\)-coordinate, substitute \(x = 8\) into the simplified function \(f(x)=x + 8\). So \(f(8)=8 + 8 = 16\). Thus, the removable discontinuity is at \((8,16)\).

Step4: Analyze the graph

The function \(f(x)=x + 8\) (with a hole at \((8,16)\)) is a linear function with slope \(1\) and \(y\)-intercept \(8\). So the graph should be a straight line with a hole at \((8,16)\). Looking at the options, we need to find the graph that represents \(y=x + 8\) with a hole at \((8,16)\).

Answer:

a. The coordinates of the removable discontinuity is \((8,16)\)
b. (We need to identify the correct graph, but based on the analysis, the graph should be a line \(y = x+8\) with a hole at \((8,16)\). Among the given options, we would look for the linear graph with a hole at \((8,16)\). For example, if one of the graphs is a line \(y=x + 8\) with an open circle at \((8,16)\), that would be the correct one. Since the options are labeled A - H, we assume the correct graph is the one that shows the line \(y=x + 8\) with a hole at \((8,16)\).)