Question

Which graph represents the function f(x)=1/x+3-2

Answer 1

Answer:

The graph as shown below.

Step-by-step explanation:

Given : The function $f(x)=\frac{1}{x-3}-2$

We have to plot the graph for the given function   $f(x)=\frac{1}{x-3}-2$  

Consider the given function $f(x)=\frac{1}{x-3}-2$

Domain of function $f(x)=\frac{1}{x-3}-2$

DOMAIN is set of input values for which the function is real and has defined values.        

So, The given function is undefined at x = 3

So, Domain is $x3$

RANGE is the set of values of dependent variable  for which the function is defined.

Inverse of given function is  $y=\frac{3x+7}{x+2}$

Now, domain of inverse function is $f\left(x\right)-2$

Now, x intercept and y- intercepts

x intercept where y = 0 and y- intercept where x= 0

Let f(x) = y

Then $y=\frac{1}{x-3}-2$

Put x = 0

thus y- intercept is $\left(0,\:-\frac{7}{3}\right)$

Now put y = 0

Then  x- intercept is $\left(\frac{7}{2},\:0\right)$

Now, Calculate the vertical and horizontal asymptotes,

Vertical asymptotes,

Go over every undefined point and check if at least one of the following statements is satisfied.

$\lim _{x\to a^-}f\left(x\right)=\pm \infty$

$\lim _{x\to a^+}f\left(x\right)=\pm \infty$

Thus, The vertical asymptotes is x = 3

And For horizontal asymptotes,

$\mathrm{Check\:if\:at\:}x\to \pm \infty \mathrm{\:the\:function\:}y=\frac{1}{x-3}-2\mathrm{\:behaves\:as\:a\:line,\:}y=mx+b$

We have y = -2 as horizontal asymptotes.

Plot we get the graph as shown below.

Answer 2

files are attached below