Question

For f(x)=3x+1 and g(x)=x squared - 6 find (f- g)(x)

Answer 1

It helps to first clarify that the notation (f - g)(x) simply means f(x) - g(x). Given that, let's look at our f(x) and our g(x) here, and use their definitions to find their difference.


`f(x)=3x+1\\g(x)=x^2-6`

When we're taking (f - g)(x), we simply substitute the expression 3x + 1 for f(x) and the expression x² - 6 for g(x) to obtain:


`(3x+1)-(x^2-6)=3x+1-x^2+6`

Or, ordering the polynomial from highest power to lowest and combining the constants:


`-x^2+3x+7`

Edit: By request, here's what would happen if you had something instead like:


`(f* g)(x)`

In this case, you'd have to *multiply* the two function expressions together. Here's what that would look like:


`(3x+1)(x^2-6)`

Using the distributive property, we can distribute the expression
`3x+1` to the terms
`x^(2)` and
`-6`:


`(3x+1)x^2-(3x+1)6`

Distributing again, we get:


`3x(x^2)+x^2-3x(6)+6=3x^3+x^2-18x+6`

And we're done.