explain how the graph of the function ( f(x)=\frac{1}{x + 8} ) can be obtained from the graph of (…

explain how the graph of the function ( f(x)=\frac{1}{x + 8} ) can be obtained from the graph of ( y=\frac{1}{x} ). then graph ( f ) and give the (a) domain and (b) range. determine the largest open intervals of the domain over which the function is (c) increasing or (d) decreasing. to obtain the graph of ( f ), shift the graph of ( y=\frac{1}{x} ) unit(s). graph the function ( f(x)=\frac{1}{x + 8} ). choose the correct graph below.

explain how the graph of the function ( f(x)=\frac{1}{x + 8} ) can be obtained from the graph of ( y=\frac{1}{x} ). then graph ( f ) and give the (a) domain and (b) range. determine the largest open intervals of the domain over which the function is (c) increasing or (d) decreasing. to obtain the graph of ( f ), shift the graph of ( y=\frac{1}{x} ) unit(s). graph the function ( f(x)=\frac{1}{x + 8} ). choose the correct graph below.

Answer

Explanation:

Step1: Analyze the transformation of the function

For a function (y = f(x)) and (y=f(x + h)), if (h>0), the graph is shifted (h) units to the left; if (h < 0), the graph is shifted (|h|) units to the right. For (y=\frac{1}{x}) and (f(x)=\frac{1}{x + 8}=\frac{1}{x-(-8)}), we have (h=- 8). So, to obtain the graph of (f(x)=\frac{1}{x + 8}) from the graph of (y = \frac{1}{x}), we shift the graph of (y=\frac{1}{x}) (8) units to the left.

Step2: Find the domain

The function (f(x)=\frac{1}{x + 8}) is a rational function. The denominator cannot be zero. Set (x+8\neq0), then (x\neq - 8). So the domain is ((-\infty,-8)\cup(-8,\infty)).

Step3: Find the range

Let (y=\frac{1}{x + 8}), solve for (x): (x=\frac{1}{y}-8). The denominator (y\neq0). So the range is ((-\infty,0)\cup(0,\infty)).

Step4: Determine the intervals of increase or decrease

Take the derivative of (f(x)) using the quotient rule. If (f(x)=\frac{1}{x + 8}=(x + 8)^{-1}), then (f^\prime(x)=-(x + 8)^{-2}=-\frac{1}{(x + 8)^{2}}). Since (f^\prime(x)=-\frac{1}{(x + 8)^{2}}<0) for all (x\neq - 8), the function is decreasing on ((-\infty,-8)) and ((-8,\infty))

Answer:

  • To obtain the graph of (f), shift the graph of (y = \frac{1}{x}) (8) units to the left.
  • (a) Domain: ((-\infty,-8)\cup(-8,\infty))
  • (b) Range: ((-\infty,0)\cup(0,\infty))
  • (c) The function is not increasing on any open interval of its domain.
  • (d) The function is decreasing on ((-\infty,-8)) and ((-8,\infty))