Exponential Function?

Question

a) For the expoenential function f(x)= -5^(x)-4, do each of the following:

i) Determine the domain and range
ii) Locate all intercepts
iii) Determine whether the function is increasing or decreasing
iv) Find any asymptotes
v) Sketch the graph of the function labeling the axes and all important points

b) For the functions f(x)= 3x^3-4x+5 and g(x)=x^2-2x+1, h= x+2, find:

i) (fog)(-5)
ii) (gof)(-5)
iii) (hogof)(1)

Answer 1

a) Given function is f(x) = -5^(x) - 4

Quadratic domain is any real value of x and range is any real value of y, lesser or equal to -5.

Horizontal asymptote is y = -4

Vertical asymptote is x = 1.8

 

 

Answer 2

b) Given functions are f(x)= 3x^3 - 4x + 5 g(x) = x^2 - 2x + 1 and h(x) =  x+2

   i) (fog)(-5) = f(g(-5))

                      = f( (-5)^2 - 2(-5) + 1)             [Since g(x) = x^2 - 2x + 1]

                      = f( 25 + 10 + 1)

                      = f(36)

                      = 3(36)^3 - 4(36) + 5              [Since f(x)= 3x^3 - 4x + 5]

                      = 3(46656) - 144 + 5

                      = 186624 - 144 + 5

                      = 186629 - 144

                      = 186485

  Therefore (fog)(-5) = 186485

 ii) (gof)(-5) = g(f(-5))

                     = g( 3(-5)^3 - 4(-5) + 5 )          [Since f(x)= 3x^3 - 4x + 5]

                     = g( 3(-125) +20 + 5 )

                     = g( -375 + 35 )

                     = g(-350)

                     = (-350)^2 - 2(-350) + 1          [Since g(x) = x^2 - 2x + 1]

                     = 122500 + 700 + 1

                     = 123201

  Therefore (gof)(-5) = 123201

 iii) (hogof)(1) = hog(f(1))

                         = hog( 3(1)^3 - 4(1) + 5 )    [Since f(x)= 3x^3 - 4x + 5]

                         = hog( 3 - 4 + 5 )

                         = hog(4)

                         = h(g(4))

                         = h( (4)^2 - 2(4) + 1)           [Since g(x) = x^2 - 2x + 1]

                         = h(16 - 8 + 1)

                         = h(17-8)

                         = h(9)

                         = 9 + 2                                [Since h(x) =  x+2]

                         = 11

  Therefore (hogof)(1) = 11

Comments

(i). (fog)(-5) = = 139829

Answer 3

(b).

The functions are f(x) = 3x³-4x+5 and g(x)=x²-2x+1 and h(x)=x+2.

(i). (fog)(-5) = f[g(-5)]

                   = f[(-5)²-2(-5)+1]             [ since g(x) = x²-2x+1]

                   = f[25+10+1]

                   = f(36)

                   = 3(36)³-4(36)+5             [ since f(x) = 3x³-4x+5]

                   = 139968-144+5

                   = 139829.