explain how the graph of the function f(x) = 7/x can be obtained from the graph of y = 1/x. then graph f and…

Explanation:

Step1: Recall function - transformation rules

For a function (y = f(x)) and (y=af(x)), if (a>0), it is a vertical - stretch or compression. Here, (f(x)=\frac{7}{x}=7\times\frac{1}{x}), and (a = 7).

Step2: Determine the transformation

To obtain the graph of (y=\frac{7}{x}) from the graph of (y = \frac{1}{x}), we vertically stretch the graph of (y=\frac{1}{x}) by a factor of 7.

For the domain and range: The function (y=\frac{7}{x}) is a rational function. The denominator cannot be zero.

Step3: Find the domain

The domain of (y = \frac{7}{x}) is all real numbers except (x = 0), which can be written in interval notation as ((-\infty,0)\cup(0,\infty)).

Step4: Find the range

The range of (y=\frac{7}{x}) is all real numbers except (y = 0), which can be written in interval notation as ((-\infty,0)\cup(0,\infty)).

To find where the function is increasing or decreasing: We can take the derivative of (y=\frac{7}{x}=7x^{-1}). Using the power - rule ((x^n)^\prime=nx^{n - 1}), we have (y^\prime=-7x^{-2}=-\frac{7}{x^{2}}). Since (y^\prime=-\frac{7}{x^{2}}<0) for all (x\neq0).

Step5: Determine increasing and decreasing intervals

The function (y = \frac{7}{x}) is decreasing on the intervals ((-\infty,0)) and ((0,\infty)) and has no intervals where it is increasing.

Answer:

To obtain the graph of (f), vertically stretch the graph of (y=\frac{1}{x}) by a factor of 7. (a) Domain: ((-\infty,0)\cup(0,\infty)) (b) Range: ((-\infty,0)\cup(0,\infty)) (c) Increasing intervals: None (d) Decreasing intervals: ((-\infty,0)) and ((0,\infty))