Question

Step by step explanations for the following

g(x)=4x-5
f(x)= -2x-3
find (f•g) (-8)
and problems on the paper numbers 5-8 please ​

Answer 1

5.
`\bold{(g \circ f)(-3) = 16}}`

6.
`\bold{h(h(-3x)) = -12x + 15 }}`

7.
`{\bold{f\left(g\left(x-3\right)\right):\quad 3x-21}}`

8.
`\bold{\text \bold g(f(-3x)) = -36x + 15}}`

Explanation:

These are examples of composite functions.

What is a composite function?
A function by definition is a process that takes a collection of inputs and produces a corresponding collection of outputs in such a way that the process produces one and only one output value for any single input value.

Since we consider a function a process it makes sense to think of two functions acting in sequence to produce a single output. The output from one function is fed into the second function.

• A composite function can be written as:
  
  `(f \circ g)(x)`

• The above can also be written as
  `f(g(x)`

• This means the output of function
  `g` is fed as input to function
  `f`

• We refer to g as the inner function and f as the outer function

• The evaluation goes from inner to outer so first
  `g(x)` is evaluated and the result fed to
  `f(x)` as input which produces the composite result

Note that we can have
`f`
`g` as the outer function, in which case the notation will be
`(g \circ f)(x)` or
`g(f(x))`

Problem Solution

5.
`g(x) = x + 3 \;\text and\; f(x) = 3x + 4`
Find
`(g \circ f)(-3) = g(f(-3))`

• First compute f(-3) in
  `f(x) = 3x + 4`
  ⇒ 3(3) + 4 = 9 + 4 = 13
  
• Then use this value for
  `x` in
  `g(x) = x + 3`
  ⇒ 13 + 3 = 16
  
• 
  `{\boxed{\bold{(g \circ f)(-3) = 16}}`

6.
`h(x) = 2x + 5`
Find
`h(h(-3x))`

• Here the output of h is fed into h again but the process of evaluation is the same
• h(-3x) in h(x) = 2x + 5
  Substitute -3x wherever you see an x
  => 2 (-3x) + 5 => -6x + 5
• Feed this into h again
  h((-3x)) = h(-6x + 5)
  = 2(-6x + 5) + 5
  = -12x +10 + 5
  = -12x + 15
  
• 
  `\\\boxed{\bold{h(h(-3x)) = -12x + 15 }}`
  

`7.\;\;\;g\left(x\right)\:=\:x-\:3,\:f\left(x\right)\:=\:3x\:-\:3\\\\\text{Find } \:f\left(g\left(x-3\right)\right)`

• 
  `\mathrm{For}\:g=x-3\:\mathrm{substitute}\:x\:\mathrm{with}\:x-3`
  
  `=x-3-3 = x - 6`
  
• 
  `\mathrm{For}\:f=3x-3\:\mathrm{substitute}\:x\:\mathrm{with}\:x-6`
  
  `=3\left(x-6\right)-3`
  
• Simplify to get
  
  `3x-21`
  
• 
  `\boxed{\bold{f\left(g\left(x-3\right)\right):\quad 3x-21}}`

.
`\bold{8.\;\;g(x) = 3x + 3, f(x) = 4x + 4}\\\\\bold{\text{Find } g(f(-3x))}`

• 
  `\text {For}\:f(x) =4x+4\:\mathrm{substitute}\:x\:\mathrm{with}\:-3x}`
  
  `=4\left(-3x\right)+4 =-12x+4`
• 
  `\mathrm{For}\:g=3x+3\:\mathrm{substitute}\:x\:\mathrm{with}\:-12x+4`
  
  `=3\left(-12x+4\right)+3 = -36x+12 + 3 = -36x + 15`
  
• 
  `\boxed{\bold{\text g(f(-3x)) = -36x + 15}}`