Sketch the graph of the function
f(x)
1
x+3

◼
  • Consider
    1
    x+3
    0
    . Impossible! So there is no
    x
    -intercept.
    • ◼
  • Now,
    f(0)
    1
    3
    . Thus the
    y
    -intercept is at
    y
    1
    3
    .
    • ◼
  • The domain of the function is
    D
    f
    (-3,+∞)
    • In[]:=
    FunctionDomain[1/Sqrt[x+3],x]
    Out[]=
    x>-3
    ◼
  • The range of the function is
    f(D)(0,+∞)
    • In[]:=
    FunctionRange[1/Sqrt[x+3],x,y]
    Out[]=
    y>0
    ◼
  • There is vertical asymptote at
    x-3
    and horizontal at
    y0
    • In[]:=
    ResourceFunction["Asymptotes"][1/Sqrt[x+3],x,y]
    Out[]=
    Horizontal{{y0,x±∞}},Vertical{{y∞,x-3}}
    In[]:=
    f[x_]:=1/Sqrt[x+3];
    In[]:=
    Plot[{f[x],0},{x,-10,5},PlotRange{{-5,5},{-2,5}},GridLines{{{-3,Directive[Thick,Red]}},None},PlotStyle{Thick,{Green,Thick,Dashed}},PlotLabelsAutomatic,AxesLabel{"x","y"}]
    Out[]=