Question

2.6.23

Let f(x) = 5x + 4 and g(x) = 4x-5. Find (f+g)(x). (f-9)(x). (fg)(x),

(x), (fog)(x), and (gof)(x). Give the domain of each.

(*+9)(x) = (Simplify your answer.)

Choch Answer

Answer 1

Check below

Explanation:

Hi there. These are operations with functions. So Let's work with these functions.

`f(x)=5x+4\\g(x)=4x-5\\`

1.
`\mathbf{a})\:(f+g)(x)=f(x)+g(x)`

`f+g(x)=5x+4 +4x-5 =\mathbf{9x-1}`

`\mathbf{b)} \:(f-g)(x)=5x+4 -4x+5=x+9\\(f-g)(x)=\mathbf{x+9}`

`\mathbf{c)} fg(x)=f(x)g(x)\\ fg(x)=(5x+4)(4x-5)=20x^(2)-25+16x-20=\mathbf{20x^(2)+16x-45}`

`\mathbf{d)} (fog)(x)=5(4x-5)+4 =20x-25+4\\(fog)(x)=\mathbf{20x-21}\\\\\ \mathbf{e)} (gof)(x)=4(5x+4)-5=20x+16-5=\mathbf{20x+11}`

2) Domain

a) The Domain of these functions is defined as the intersection of the first function's Domain f(x) and g(x)'s domain:

`A \cap B\\Domain_A=(-\infty \:<x<\infty \:)\\Domain_B=(-\infty \:<x<\infty \:)\\`

So the Domain of (f+g)(x),(f-g), and (fg)(x): Real set.

The functions have no discontinuity, nor restrictions.