Question

Can you help me I have no idea how to do this

Answer 1

`(f + g)(x) = 7 - 6x` --- (1)

`(f - g)(x) = 6x - 1` -- (2)

`(f . g)(x) =4(5^x)` ---- (3)

`(f / g)(x) = (4)/(5^x)` ----- (4)

`(f + g)(x) = -7x-7` --- (5)

`(f - g)(x) = 9x+ 9` --- (6)

`(f . g)(x) = (-8)(x+1)^2` --- (7)

`(f / g)(x) = -(1)/(8)` ----- (8)

`F(t) = (9)/(5)t^2 + 32` ---- (9)

`T(t) = (t - 2)^2 - 4` --- (10)

Explanation:

Given

`f(x) = 3`

`g(x) = 4 - 6x`

Solving (1): (f + g)(x)

`(f + g)(x) = f(x) + g(x)`

So, we have:

`(f + g)(x) = 3 + 4 - 6x`

`(f + g)(x) = 7 - 6x`

Solving (2): (f - g)(x)

`(f - g)(x) =f(x) - g(x)`

So, we have:

`(f - g)(x) =3 - (4 - 6x)`

`(f - g)(x) =3 - 4 + 6x`

`(f - g)(x) =-1 + 6x`

`(f - g)(x) = 6x - 1`

Given

`f(x) = 4`

`g(x) = 5^x`

Solving (3): (f . g)(x)

`(f . g)(x) =f(x) * g(x)`

So, we have:

`(f . g)(x) =4 * 5^x`

`(f . g)(x) =4(5^x)`

Solving (4): (f / g)(x)

`(f / g)(x) = (f(x))/(g(x))`

So, we have:

`(f / g)(x) = (4)/(5^x)`

Given

f(x) = x + 1

g(x) = -8 - 8x

Solving (5): (f + g)(x)

`(f + g)(x) = f(x) + g(x)`

So, we have:

`(f + g)(x) = x + 1 -8-8x`

Collect Like Terms

`(f + g)(x) = x -8x+ 1 -8`

`(f + g)(x) = -7x-7`

Solving (6): (f - g)(x)

`(f - g)(x) = f(x) - g(x)`

So, we have:

`(f - g)(x) = x + 1 -( -8-8x)`

`(f - g)(x) = x + 1 +8+8x`

Collect Like Terms

`(f - g)(x) = x +8x+ 1 +8`

`(f - g)(x) = 9x+ 9`

Solving (7): (f . g)(x)

`(f . g)(x) = f(x) . g(x)`

So, we have:

`(f . g)(x) = (x+1) . (-8 - 8x)`

Factorize

`(f . g)(x) = (x+1) .(-8) (1 + x)`

Rewrite as:

`(f . g)(x) = (x+1) .(-8) (x+1)`

`(f . g)(x) = (-8)(x+1) (x+1)`

`(f . g)(x) = (-8)(x+1)^2`

Solving (8): (f / g)(x)

`(f / g)(x) = (f(x))/(g(x))`

So, we have:

`(f / g)(x) = ((x+1))/((-8-8x))`

Factorize

`(f / g)(x) = ((x+1))/(-8(1+x))`

`(f / g)(x) = (1)/(-8)`

`(f / g)(x) = -(1)/(8)`

Solving (9):

From the question, we have that:

`F(c) = (9)/(5)c + 32`

`C(t) = t^2`

Required

Determine function F in terms of c

The implication of this question is to solve for
`F(c(t))`

If
`F(c) = (9)/(5)c + 32` and
`C(t) = t^2`,

Then

`F(c(t)) = (9)/(5)*t^2 + 32`

`F(c(t)) = (9)/(5)t^2 + 32`

This can be rewritten as:

`F(t) = (9)/(5)t^2 + 32`

Solving (10):

`T(h) = h^2 - 4`

`h(t) = t - 2`

Required

Find
`T(h(t))`

If
`T(h) = h^2 - 4` and
`h(t) = t - 2`, then

`T(h(t)) = (t - 2)^2 - 4`

This is gotten by substituting t -2 for h

The solution can be rewritten as:

`T(t) = (t - 2)^2 - 4`